sensors and actuators introduction to sensors · 5 potentiometer resistance of a material given by...

Department of Electrical Engineering

Electronic Systems

Sensors and Actuators Introduction to sensors

Sander Stuijk

([email protected])

2

DISPLACEMENT SENSORS (Chapter 7.1)

3 Applications

4 Sensor classification

sensor type quantity physical effect sensor

resistive displacement resistance potentiometer

inductive

displacement reluctance variable differential

transformer

inductive displacement Joule effect, Villari effect,

Wiedemann effect, Matteucci

effect

magnetostrictive

electromagnetic velocity Hall effect Hall effect

electromagnetic velocity Faraday’s law linear velocity

capacitive acceleration capacitance variable capacitor

capacitive acceleration capacitance differential capacitor

5 Potentiometer

resistance of a material given by

ρ – resistivity of material

l – length

A – cross-sectional area of material

change in the length changes the resistance of the material

A

lR

potentiometer (pot) is a variable resistor

pot wiper mechanically coupled to object

whose displacement must be measured

movement (linear or rotation) of object

causes change in resistance

6 Potentiometer

output voltage proportional to displacement

output voltage does not depend on the resistance RT

potentiometer is a ratio metric device

Vr

(1-α)RT

αRT vo dD

rr

T

Tr

T

To V

D

dV

DR

dRV

R

Rv

rr

T

Tr

TT

To VV

R

RV

RR

Rv

1

7 Potentiometer


what happens when a load Rm is connected to the sensor?

Thévenin equivalent circuit

open circuit voltage

output resistance

voltage across load

rr

T

Tr

TT

To VV

R

RV

RR

Rv

1

Vr

(1-α)RT

αRT Rmvm

Ro

Rmvovm

rr

T

To VV

R

Rv

T

TT

TTo R

RR

RRR

1

1

1

TTo RRR 1//

r

Tm

mo

om

mm V

RR

Rv

RR

Rv

1

8 Potentiometer


what happens when a load Rm is connected to the sensor?

voltage across load

sensor becomes linear when k >> 1(Rm >> RT)

relative error of the sensor

relative error depends on position of the wiper (i.e. it depends on α)

no error at the ends of the scale

rr

T

Tr

TT

To VV

R

RV

RR

Rv

1

r

Tm

mm V

RR

Rv

1

Tm RRk rm V

k

v

11

1

1

1

1

1

1

11

kkk

kk

V

VVk

k

v

vv

r

rr

o

om

9 Potentiometer

when is the maximal relative error reached?

maximal error reached when

maximal error

increasing load resistor decreases error

error is symmetrical around the center

0

d

d

21

211211

1

1

k

k

kd

d

d

d

21

21

k

k

d

d

2

1021

2v

d

dvu

d

duv

v

u

d

d

recall that

α

ε

k=1 25.0

25.0

1

15.0max

kk

10 Potentiometer – loading error

relative error (loading error) can be decreased by

increasing load resistor Rm

placing a resistor equal to Rm at the top side of the potentiometer

what is the output voltage of the sensor?

two options to compute output voltage

use Thévenin equivalent circuit

output voltage is voltage divider of two pair of parallel resistors

with

Vr

(1-α)RT

αRT Rmvm

Rm

r

xy

y

m VRR

Rv

mT

mTmTx

RR

RRRRR

1

1//1

mT

mTmTy

RR

RRRRR

//





what is the output voltage of the sensor?

output voltage at α = 0.5

Vr

(1-α)RT

αRT Rmvm

Rm

r

xy

y

m VRR

Rv

T

m

R

Rk

r

mT

TTmm V

RR

RRRv

12 rm V

k

kv

12

1

rrm VVk

kv

2

1

5.0

5.05.05.0

additional resistor Rm enforces

vm = Vr/2 at central position

zero error at this point


example – set zero error at any desired point

which value must R1 and R2 have such that a wiper displacement

of ±15% of its stroke around the position corresponding to ¼ of

the full-scale value produces a change in voltage of only 10% with

respect to the full-scale voltage?

use a = RT/R1 and b = RT/R2

Vr

(1-α)RT

αRT R2va

R1

r

TT

Ta V

RRRR

RRv

1////

//

12

2

r

T

T

T

T

T

T

V

RR

RR

RR

RR

RR

RR

1

1

1

1

2

2

2

2

r

TTTT

TT VRRRRRRRR

RRRR

2112

12

11

1

rm V

ba

av

1111

11

rm V

ba

av

11

11


example – set zero error at any desired point

which value must R1 and R2 have such that a wiper displacement

of ±15% of its stroke around the position corresponding to ¼ of

the full-scale value produces a change in voltage of only 10% with

respect to the full-scale voltage?

two constraints

1) at α = 0.25 + 0.15 it must hold that va = (0.25 + 0.05)Vr

2) at α = 0.25 - 0.15 it must hold that va = (0.25 - 0.05)Vr

constraints imply

solving constraints yields

this implies

Vr

(1-α)RT

αRT R2va

R1

ba

a

625

1063.0

ba

a

990

1092.0

5.3a 6.9b

TRR 29.01 TRR 10.02





using a symmetrical voltage supply

what is the output voltage of this sensor?

use Kirchhoff current law

solving this yields

error is zero at end of the wiper (α = 0, α = 1)

error is zero at central position (α = 0.5)

Vr (1-α)RT

αRT

Rm

vmVr

T

mr

T

mr

m

m

R

vV

R

vV

R

v

10

k

Vv r

m

11

12

15 Potentiometer – lead-wire resistance

3-wire circuit output voltage

assume k >> 1, hence loading error is negligible

output at α = 0

output at α = 1

circuit has two errors

reduced sensitivity

offset voltage

due to current

through Rw3

(1-α)RT

αRT

Rw1

Vr

vm

Rw2

Rw3

r

wwT

wm V

RRR

Rv

31

30

r

wwT

wTm V

RRR

RRv

31

31

1

10α

vm/Vr ideal output

real output

16 Potentiometer – lead-wire resistance

4-wire circuit output voltage

assume k >> 1, hence loading error is negligible

output at α = 0

output at α = 1

no offset voltage

sensitivity equal to 3-wire circuit

00 mv

r

wwT

Tm V

RRR

Rv

41

1

(1-α)RT

αRT

Rw1

Vr

vm

Rw2

Rw3

Rw4

1

10α

vm/Vr ideal output

real output

17 Potentiometer – construction

construction: wire wound around a coil

problem: wiper makes contact with 1 or 2 wires

results in uneven voltage steps

sensor has a variable resolution

resolution: smallest change in input

stimulus which can be sensed

consider average resolution when

computing displacement

construction: high-quality resistive film

theoretically an infinitesimal resolution

resolution limited by

noise in circuit

non-uniformity of material

18 Potentiometer

pro’s and con’s of potentiometers

(+) cheap

(+) simple concept

(+) no complicated processing circuits needed

(-) physical coupling to object needed

noticeable mechanical load

wear on the device

friction causes heating error

(-) excitation voltage causes heating error

(-) limited resolution

(-) substantial non-linearity error

sensors and actuators introduction to sensors · 5 potentiometer resistance of a material given by...

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