A. Static conditions describe how sensors output static relationships between input and output values. This includes the static characteristic function f(x) relating input x to output y, and the inverse function F. Accuracy considerations include sensitivity, resolution, error sources like offset and gain errors. B. Dynamic conditions describe how sensors respond to changing input values over time. This includes responses that are first or second order systems with properties like rise time and time constant. Common sensor types are discussed like thermistors that have sensitivity that varies with temperature. C. Specific sensors are briefly described like the LM92 temperature sensor, KTY81 thermistor, infrared distance sensor GP2Y0A02YK0F, and pressure

1. 1
Sensors Characteristics

2. 2
A. Static conditions 0
,
0 

dt
dy
dt
dx
B. Dynamic conditions )
(
))
(
(
0
t
x
dt
t
y
d
a
k
k
k 




3. 3
A.1 Static characteristics
)
(x
f
y 
Sensors
x
y
Ko
x
K
y 

 K - scaling factor in linear systems
)
(x
f - polynomial, exponential, logarithmic function
or experimental determination
)
(x
f -result from modeling of the system (sensor)
-univocal process

4. 4
A.1 Static characteristic
A.2 Invers characteristic (function)
Sensors
x
y
- Perturbative parameters like: humidity, temperature, altitude, ….
i

)
,..
,
( 1 n
x
f
y 


i

F = f -1
Sensors
x
y
f
F
]
[
);
1
(
])
[
( 0
0




 T
R
C
T
R 
]
[
;
)
(
])
[
( 0
0
0 C
R
R
R
R
T





f:
F:

5. 5
]
[
]);
[
],
[
( 1 Hz
C
T
mm
x
f
y o


F = f -1
 
25
02
.
0
1000
1000
0106
.
0
]
[ 1
2
0
2













 
y
y
mm
x
])
[
(
1 
 R


A.1 Static characteristic of complex sensors-Vibrating Wire Jointmeter
A.2 Invers characteristic (function) Vibrating Wire Jointmeter

6. 6
Static conditions
A.1, A.2 e.g.
A.1 Thermistor 1k A.2
A.1 Characteristic processing, linearization
   
   















max
max
max
max
min
1
min
1
2
1
2
1
1
;
;
,
,
:
...
;
;
,
,
:
y
y
x
x
y
y
x
x
y
y
x
x
y
y
x
x
n
n
n
n
n
]
,
[
]
,
[
: max
min
max
min y
y
x
x 

0
)
(
)
,
( 



 y
x
y
E
Approximation function of
experimental data

7. 7
Static conditions, example
A.1 GP2Y0A02YK0F A.2
]
55
.
1
8
.
2
[ 

out
V
]
8
.
0
55
.
1
( 

out
V
]
45
.
0
8
.
0
( 

out
V
Γ2(Vout)= 119.22 - 52.40 ∙ Vout;
Γ1(Vout)=71.31-20.36∙ Vout ;
Γ3(Vout)= 223.88-183.19∙ Vout

8. 8
Static conditions
A.1, A.2 e.g. GP2Y0A02YK0F
L
L
E C
C
C 
 3
,
2
,
1
L _ BSLF - Best Straight Line Fit
-Mathematical determination
- Software consideration (Curve expert)

9. 9
Static conditions
A.3 Full Scale input range
Out of range - recalibration, saturate,
- new calibration function
A.4 Full Scale output range
[xmin…xmax]
GP2Y0A02YK0F
)
(x
f
y 
)
(x
f
y  ]
[ max
min y
y
y 

]
45
.
0
8
.
2
[ 

out
V
GP2Y0A02YK0F
cm
L ]
150
15
[ 

ymin  0 for xmin = 0, and: ymin = 0 when xmin  0.
x
y f1
f2
f3
f4
X min X max
Y min
y max

10. 10
Static conditions
A.5 Sensitivity
Relative sensitivity
1
2
1
2
x
x
y
y
S



min
max
min
max
x
x
y
y
S


 Linear system, K=S
If “K” is higher is better,
but the range is lower.
i
i
x
x
dx
dy
S

 Non-linear system
n
x
m
y
II 

 2
:
order x
m
dx
dy
S 


 2











[x]
%
;
2
1
m
dx
dy
x
x
S
X

]
[
);
1
(
])
[
(
:
order 0
0




 T
R
C
T
R
I  


 0
R
dT
dR
S 




 
C
0

11. 11
Static conditions
A.5 Sensitivity with external parameters
A.6 Resolution
- sensorial systems with digital unit
- depend by of ADC’s number of bits
)
,..
,
( 1 n
x
f
y 

  







i
i
f
x
x
f
y 
 i
f



- Parasitic sensitivity
T
K
l
K
T
S
l
l
T
S
T
S
l
R 






























 '
)
1
(
1
)
1
( 0
0
0
0 





   
C
K
m
K 



12. 12
Static conditions
A.5 Error-Accuracy
- function approximation,
- Noise,
- Improper device,…
;
,
x x
X
X 



 
 x
X 

 X estimation of x - true value
100
[%]
or
100
[%] 





x
x
X
x
x
X

 ;
0

x
[%]
100




x
X
R
   ];
[
10
then,
10 6
%
6
% ppm
if ppm 

 




13. 13
Static conditions
A.5.1 Offset Error
y’= f’(x)= f(x)+εoff
A.5.2 Gain Error
y = f(x) linear system
off
x
f 

 )
0
(
' )
0
(
'
)
( 


 x
f
x
f
y off

ideal
min
max
min
max
x
x
y
k
y



real
min
max
min
max '
'
'
x
x
y
k
y


 k
r
y
r
y
C
C
x
x
y
x
x
y
k 







min
max
min
max
min
max
min
max
)
'
'
(
'
, true

14. 14
Static conditions
A.5.3 Non-linear Error
-Caused by approximation
A.5.6 Repeatability Error
..
3
3
2
2
)
(
1
0
)
(
1
0
0
)
(
x
k
x
k
x
g
x
k
k
x
g
x
k
k
i
i
x
i
k
x
f















[%]
100


 R
R



15. 15
Problemes
A given sensor has a specified linearity error of 1 % of the
reading plus 0.1 % of the full-scale output (FSO). A second
sensor having the same measurement range has a specified
error of 0.5 % of the reading plus 0.2 % FSO. For what range
of values is the first sensor more accurate than the second
one? If the second sensor had a measurement range twice
that of the first one, for what range of values would it be the
more accurate?

16. 16
Dinamic conditions
)
(
))
(
(
0
t
x
dt
t
y
d
a
k
k
k 



 
pol
s
a
s
a
s
a
s
X
s
Y
s
H
s
X
s
Y
a
s
a
s
a
t
y
a
dt
t
y
d
a
dt
t
y
d
a
L
k
k
k
k
k
k

























,
j
s
;
...
1
)
(
)
(
)
(
)
(
)
(
...
)
(
))
(
(
...
))
(
(
0
1
0
1
1
1
1


Exponential Growth
Oscilation
Damped oscillator
Over Damped

17. 17
Dynamic conditions
 Potentiometer
 Thermistor
)
(
)
( 0 t
x
a
t
y 

)
(
)
(
))
(
(
0
1 t
x
t
y
a
dt
t
y
d
a 



y
sensitivit
static
1
constant,
time
1
)
(
)
(
)
(
)
(
1
)
(
))
(
(
0
0
1
0
0
1
a
k
a
a
s
k
s
X
s
Y
s
H
t
x
a
t
y
dt
t
y
d
a
a












),
(
1
)
( s
X
s
k
s
Y 













 1
1
1
)
(
s
k
s
k
s
s
k
s
Y  
)
(
)
( 1
s
Y
L
t
y 

)
1
(
)
( 
t
e
k
t
y




signal
step
unit
/
1
))
(
(
)
( s
t
x
L
s
X
if 

zero order
firs order

18. 18
Dynamic conditions
Rising time ( tC )
Response time (tt)
Signal rise
(εM)
)
(
)
(
))
(
(
))
(
(
0
1
2
2
2 t
x
t
y
a
dt
t
y
d
a
dt
t
y
d
a 




 second order

19. 19
LM92 temperature sensor

20. 20
KTY81

21. 21
GP2Y0A02YK0F

22. 22
MPX53

23. 23
S18U