sensors and actuators · mean time between collisions (τ) resistor as temperature sensor some...
TRANSCRIPT
Department of Electrical Engineering
Electronic Systems
Sensors and Actuators Sensor Physics
Sander Stuijk
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 is the maximal deviation from the linear transfer function
nonlinearity must be deduced from the actual transfer function or
from a calibration curve
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
nonlinearity is the maximal deviation from the linear transfer function
nonlinearity must be deduced from the actual transfer function or
from a calibration curve
~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
errors are deviations from the “ideal”
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
accuracy may be represented
in terms of measured value (Δ)
in percent of full scale input (%)
in terms of output signal (δ)
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 may be represented
in terms of measured value (Δ)
in percent of full scale input (%)
in terms of output signal (δ)
accuracy of the LM135
measured value: ±1°C
percentage: ±1°C/(150°C)∙100 =
±0.7%
output: ±10mV
19 Errors
errors are deviations from the “ideal”
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
errors are deviations from the “ideal”
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
general form linear differential equation
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
42 Zero-order system
general form linear differential equation
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
43 Zero-order system
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
45 First-order system
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
46 First-order system
, 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
Example – liquid-in-glass thermometer
conservation of energy provides relation between
fluid temperature (Tf) and liquid temperature (Ti)
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
Example – liquid-in-glass thermometer
conservation of energy provides relation between
fluid temperature (Tf) and liquid temperature (Ti)
relation between liquid level (xo) and
liquid temperature (Ti)
what are sensitivity (k) and time constant (τ)?
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
Example – liquid-in-glass thermometer
what are sensitivity (k) and time constant (τ)?
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
Example – liquid-in-glass thermometer
what are sensitivity (k) and time constant (τ)?
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