sensors and actuators · mean time between collisions (τ) resistor as temperature sensor some...

Department of Electrical Engineering

Electronic Systems

Sensors and Actuators Sensor Physics

Sander Stuijk

([email protected])

4

SENSOR CHARACTERISTICS (Chapter 2)

5 Resistance

resistance of a material is defined as

resistance depends on geometrical factors

length of wire (l)

cross-sectional area (a)

resistance depends on temperature

number of free electrons (n)

mean time between collisions (τ)

resistor as temperature sensor

some types have almost linear

relation between temperature t (°C)

and resistance R (Ω)

example: platinum (PT100) sensor

i

VR

a

l

ne

m

a

lR

2

a

lR

tRRt

4

0 1008.391

6 Transfer function

sensors translates input signal to electrical signal

transfer function gives relation between input and output signal

V1 (5V)R5 (3.01kΩ)

Rt, R0 = 100 Ω Vout

1

5

VRR

RV

t

tout

tRRt

4

0 1008.391

7 Signal processing

43

4

RR

RVV out

52

21

25

5

//

//

//

//

RRR

RRV

RRR

RRVV

t

t

t

tout

V1 (5V)

R5 (3.01 kΩ)

Rt

Vout

R1 (11 kΩ)R2 (11.8 kΩ)

R3 (105 kΩ)

R4 (12.4 kΩ)

+

-

25

5

34

4

52

21

//

//

//

//

RRR

RR

RR

R

RRR

RRV

VVV

t

t

t

t

out

8

temperature (°C)

voltage (

v)

Signal processing

sensitivity increased from 0.63mV/°C to 6.67mV/°C

non-linearity has also been decreased...

Vout

sensor

output

9 Nonlinearity

assumption: resistance has a linear dependency on temperature

temperature (°C)

voltage (

V)

“ideal” linear transfer function

“real” transfer function

temperature (°C)

err

or

(V)

10 Nonlinearity

assumption: resistance has a linear dependency on temperature

error can be expressed as deviation from actual temperature

temperature (°C)

err

or

(V)

temperature (°C)

err

or

(°C

)

11

temperature (°C)

err

or

(°C

) Nonlinearity

nonlinearity is the maximal deviation from the linear transfer function

nonlinearity must be deduced from the actual transfer function or

from a calibration curve

real transfer

function

ideal transfer function

nonlinearity

12




nonlinearity can be reduced with signal processing electronics

temperature (°C)

err

or

(°C

) Nonlinearity

V1 (5V)

R5 (3.01kΩ)

PT100

(100Ω)Vout

R1 (11kΩ)R2 (11.8kΩ)

R3 (105kΩ)

R4 (12.4kΩ)

+

-

raw sensor output

13




~10x reduction in nonlinearity due to signal processing electronics

temperature (°C)

err

or

(°C

) Nonlinearity

raw sensor output

temperature (°C)

err

or

(°C

)

network output

14 Errors

errors are deviations from the “ideal”

transfer function

sources

nonlinearity

materials used

construction tolerances

aging

operational errors

calibration errors

impedance matching errors

noise

....

15 Errors


transfer function

types of errors

static errors: not time dependent

dynamic errors: time dependent

systemic errors: errors are

constant at all times and

conditions

random errors: different errors in a

parameter or at different operating

times

16 Accuracy

accuracy is a bound on the maximal

deviation of the true input for any output

of the sensor

example: if the accuracy is ±3°C, the

measured temperature is the true value

±3°C

accuracy may be represented

in terms of measured value (Δ)

in percent of full scale input (%)

in terms of output signal (δ)

17 Accuracy





LM135 - precision temperature sensor

sensitivity: +10mV/°C

range: -55°C to +150°C

span: 150°C - (-55°C) = 205°C

input full scale: 150°C

output full scale: 4.2V

uncalibrated temp error: ±1°C

what is the accuracy of this sensor?

18 Accuracy





accuracy of the LM135

measured value: ±1°C

percentage: ±1°C/(150°C)∙100 =

±0.7%

output: ±10mV

19 Errors


transfer function

sources of errors (seen so far)

nonlinearity

other sources of errors

calibration errors

repeatability

hysteresis

saturation

dead band

...

20 Calibration error

calibration data is usually supplied by the manufacturer

calibration error is the inaccuracy permitted by the manufacturer

when calibrating a sensor in the factory

accurate

measurement

measurement

with error

12

1ss

aaa

12 ssb

offset error

slope error

21 Calibration and nonlinearity

nonlinearity needs to be considered when calibrating sensor

several calibration methods are used

use range points (line 1)

limit span to useful range and use these range points (line 2)

use tangent of single calibration point (line 3)

use linear best fit (line 4)

4

22 Errors


transfer function

sources of errors (seen so far)

nonlinearity

calibration errors

other sources of errors

repeatability

hysteresis

saturation

dead band

...

23 Repeatability

repeatability is the failure of a sensor to represent the same value

under identical conditions when measured at different times

source: thermal noise, buildup charge, material plasticity, ...

%100

FS

r

24 Hysteresis

hysteresis is the deviation of the sensor’s output at any given point

when approached from two different directions

caused by electrical or mechanical properties

mechanical friction

magnetization

thermal properties

loose linkages

hh

25 Example – magnetoresistive sensor

sensor can be used to measure the position of magnetic objects

resistivity of magnetoresistive sensor has relation with strength and

position of magnetic field

sensor moved along X axis

Hx provides auxiliary field

variation in Hy is a measure for the displacement

sensor output voltage V0 follows Hy curve

Hy

Hx

26 Example – magnetoresistive sensor

sensor can be used to measure the position of magnetic objects

resistivity of magnetoresistive sensor had relation with strength and

position of magnetic field

hysteresis error

too strong magnet or sensor to close to magnet

Hx exceeds maximal Hx

dipoles flip

sensor has hysteresis loop: ABCD

Hy

Hx

36 Static and dynamic characteristics

static characteristics

values given for steady state measurement

dynamic characteristics

values of the response to input changes

many sensors have a time-dependent behavior

output signal needs time to adapt to change in input

example - LM135 temperature sensor

voltage step at input

output needs time to settle

37 Dynamic error

dynamic error is the difference between the indicated value and true

value of measured quantity when static error is zero

difference in sensor response when input is constant or varies

two important aspects

magnitude of error

speed of response (delay)

different inputs considered when analyzing dynamic characteristics

step (e.g., sudden temperature change)

ramp (e.g., gradual temperature change)

sinusoid (e.g., sound waves)

any real signal can be described as superposition of these signals

38 Transfer function

input-output behavior of sensor captured with constant-coefficient

linear differential equation (sensor is linear time-invariant system)

general form linear differential equation

y(t) – output quantity

x(t) – input quantity

t – time

ai, bi – constant physical parameters of system

solution to equation can be computed using Laplace transform

transfer function of a system is defined as

)(...)()(

)()(

...)()(

01

1

10

1

11

1

1 txbdt

txdb

dt

txdbtya

dt

tyda

dt

tyda

dt

tyda

m

m

mm

m

mn

n

nn

n

n

01

1

1

01

1

1

...

...

)(

)(

asasasa

bsbsbsb

sX

sYn

n

n

n

m

m

m

m

39 Transfer function

transfer function does not capture the instantaneous ratio of time-

varying quantities

inverse Laplace transform is needed to go back to time domain

Laplace form is convenient for combining transfer functions

initial condition can be ignored in transformation when all initial

conditions are zero

this is true for many practical systems

01

1

1

01

1

1

...

...

)(

)(

asasasa

bsbsbsb

sX

sYn

n

n

n

m

m

m

m

...)()(

)()(2

2

dt

txdC

dt

tdxBtAxty

40

1

21

2

2

nn

adt

sss

kkk

Transfer function

complex sensors combine several transducers and a direct sensor

combination of transfer functions of all transducers gives transfer

function of complex sensor

1s

kt

12

2

2

nn

d

ss

k

ak

transducer direct sensor amplifier

measured

quantity voltage

sensor

measured

quantity voltage

41 Zero-order system


many systems are simpler ...

example – potentiometric displacement sensor

this system is “memory” less ro V

D

tdtv

)()(

Vr

(1-α)RT

αRT vo dD

t

t

d

vo

)(...)()(

)()(

...)()(

01

1

10

1

11

1

1 txbdt

txdb

dt

txdbtya

dt

tyda

dt

tyda

dt

tyda

m

m

mm

m

mn

n

nn

n

n

)()( txkty



many systems are simpler ...

differential equation for zero-order systems

static sensitivity given by k

note: S was used before when discussing static characteristics

S = k

zero-order system represents ideal or perfect dynamic performance

)()()()()(0

000 txktx

a

btytxbtya

)(...)()(

)()(

...)()(

01

1

10

1

11

1

1 txbdt

txdb

dt

txdbtya

dt

tyda

dt

tyda

dt

tyda

m

m

mm

m

mn

n

nn

n

n


zero-order system represents ideal or perfect dynamic performance

demonstrated with response to step at input

no dynamic error present in zero-order systems

none of the elements in the sensor stores energy

ω

ω

Ao/Ai

φ

K

step input frequency response

t

t

x(t)

y(t)

ci

kci

44 First-order system

many systems are not ideal...

(parasitic) capacitance or inductance

are often present

example – liquid-in-glass thermometer

input – temperature Ti(t) of

environment

output – displacement xo of the

thermometer fluid

liquid column has inertia (i.e.

transfer function is not ideal)

Ti(t)

Tf

xo=0

xo


first-order system contains one energy storing element

differential equation for first-order system

engineering practice to only consider x(t) and not its derivatives

solve equation to obtain transfer function

k – static sensitivity

τ – time constant

)()()(

001 txbtyadt

tdya

)()()(

0

0

0

1 txa

bty

dt

tdy

a

a

0

1

0

0 ,a

a

a

bk

)()(1 sXksYs

1)(

)(

s

k

sX

sY


, with

static input implies all derivatives are zero

static sensitivity (k) is the amount of output per unit input when the

input is static (constant)

time constant (τ) determines the lag of the output signal on a change

in the input signal

0

1

0

0 ,a

a

a

bk

1)(

)(

s

k

sX

sY

step at input response at output

t

y(t)

kci

t

x(t)

ci

small τ large τ

47

Ti(t)

Tf

xo=0

xo

Example – liquid-in-glass thermometer

conservation of energy provides relation between

fluid temperature (Tf) and liquid temperature (Ti)

Vb – volume of bulb [m3]

ρ – mass density of thermometer fluid [kg/m3]

C – specific heat of thermometer fluid [J/(kg°C)]

U – overall heat-transfer coefficient across bulb

wall [W/(m2°C)]

Ab – heat transfer area of bulb wall [m2]

ibfb

f

b TUATUAdt

dTCV

48

Ti(t)

Tf

xo=0

xo




relation between liquid level (xo) and

liquid temperature (Ti)

xo – displacement from reference mark [m]

Kex – differential expansion coefficient of fluid

and bulb [m3/(m3°C)]

Vb – volume of bulb [m3]

Ac – cross sectional area of capillary tube [m2]

what are sensitivity (k) and time constant (τ)?

ibfb

f

b TUATUAdt

dTCV

f

c

bexo T

A

VKx

49

Ti(t)

Tf

xo=0

xo




relation between liquid level (xo) and

liquid temperature (Ti)


combining equations gives differential equation

for whole system

ibfb

f

b TUATUAdt

dTCV

f

c

bexo T

A

VKx

bex

ocff

c

bexo

VK

xATT

A

VKx

ibfb

f

b TUATUAdt

dTCV

ibo

bex

cbo

ex

c TUAxVK

AUA

dt

dx

K

CA

50

Ti(t)

Tf

xo=0

xo



combining equations gives differential equation

for whole system

general first-order system

sensitivity [m/°C]

time constant [s]

ibo

bex

cbo

ex

c TUAxVK

AUA

dt

dx

K

CA

0

1

0

0 ,a

a

a

bk )()(

)(

0

0

0

1 txa

bty

dt

tdy

a

a

c

bex

A

VKk

b

b

UA

CV

51

Ti(t)

Tf

xo=0

xo



sensitivity [m/°C]

time constant [s]

sensitivity and time constant related to physical

parameters

larger sensitivity (k) requires larger bulb volume (Vb)

larger bulb volume (Vb) increases time constant (τ)

effect partially offset by increased contact area (Ab)

careful selection of parameters is required

c

bex

A

VKk

b

b

UA

CV

sensors and actuators · mean time between collisions (τ) resistor as temperature sensor some...

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