Nguyễn Công Phương

Sensors and Analytical Devices

Sensors Characteristics

Contents

A. Introduction B. Sensors Characteristics I. Static Characteristics II. Dynamic Characteristics

C. Some Basic Measurement Methods D. Measurement Systems

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Static Characteristics

Accuracy Precision Repeatability Reproducibility Stability Error Noise Drift Resolution

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Minimum detectable signal 11. Calibration curve 12. Sensitivity 13. Linearity 14. Selectivity 15. Hysteresis 16. Measurement range 17. Response & recovery time

Accuracy

• Accuracy: the correctness of a sensing system’s output in comparison to the actual value of a measurand.

• To assess the accuracy:

– The system is benchmarked against a standard measurand,

or

– The output is compared with a measurement system with a

superior accuracy

More accurate

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Less accurate

Precision

• Precision: the capacity of a sensing system to give the same reading when repetively measuring the same measurand under the same condition.

• It is a statistical parameter & can be assessed by the standard deviation (variance) of a set of readings of the system for similar inputs

Low precision Low accuracy High precision High accuracy

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Repeatability

• When all operating & environmental

conditions remain constant, repeatability is the sensing system’s ability to produce the same response for successive measurements.

• It is closely related to precision.

Temperature

The actual temperature

10:00 10:02 20:00 20:02 Time

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10:01 20:01

Reproducibility

• The sensing system’s ability to produce the

same responses after measurement conditions have been altered.

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Stability

• The sensing system’s ability to produce the

same output value when measuring the same measurand over a period of time.

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Static Characteristics

Accuracy Precision Repeatability Reproducibility Stability Error Noise Drift Resolution

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Minimum detectable signal 11. Calibration curve 12. Sensitivity 13. Linearity 14. Selectivity 15. Hysteresis 16. Measurement range 17. Response & recovery time

Error

• Error is the difference between the actual

value of the measurand and the value produced by the sensing system.

• It can be systematic or random.

Absolute error Output True value

Relative error

Output True value True value

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Noise

• Noise: the unwanted fluctuations in the output signal of the

sensing system, when the measurand is not changing. • Electronic noise: thermal energy causes charge carriers to

move in random motion, which results in random variations of current and/or voltage.

• Shot noise: the random fluctuations, which are caused by the carriers’ random arrival time, produce shot noise. • Generation – recombination noise: it is produced from the

generation & recombination of electrons & holes in semiconductors

• Pink noise (or 1/f noise): in this type of noise the

components of the frequency spectrum of the interesting signals are inversely proportional to the frequency.

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Drift

• It is observed when a gradual change in the sensing system’s output is seen, while the measurand actually remains constant.

• It is the undesired change that is unrelated to

the measurand. Temperature

The actual temperature

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Time

Resolution

• Resolution (sometimes discrimination): the minimal change of the measurand that can produce a detectable increment in the output signal.

• It is strongly limited by any noise in the signal.

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Minimum Detectable Signal

• Minimum Detectable Signal (MDS) is the minimum signal increment that can be observed, when all interfering factors are taken into account.

• When the increment is assessed from zero, the value is generally referred to as threshold or detection limit.

• If the interferences are large relative to the input, it will be difficult to extract a clear signal & small MDS can not be obtained.

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Static Characteristics

Accuracy Precision Repeatability Reproducibility Stability Error Noise Drift Resolution

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Minimum detectable signal 11. Calibration curve 12. Sensitivity 13. Linearity 14. Selectivity 15. Hysteresis 16. Measurement range 17. Response & recovery time

Calibration Curve

• Calibration curve: the relationship between the

measured variable x & the signal variable generated by the system y.

• A sensing system has to be calibrated against a known measurand to assure that the sensing results in correct outputs.

y

x

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Sensitivity

• Sensitivity: the ratio of the incremental change in the sensor’s output (Δy) to the incremental change of the measurand in input (Δx). • An ideal sensor has a large & preferably constant sensitivity in its operating range.

y

y

x

x

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Linearity

• Linearity: the closeness of the calibration

curve to a specified line.

• The degree of resemblance to a straight line

describes how linear a system is.

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Selectivity

• Selectivity: the sensing system’s ability to

measure a target measurand in the presence of other interferences.

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Hysteresis

• Hysteresis: the difference between output

readings for the same measurand, depending on the trajectory followed by the sensor. • It may cause false & inaccurate readings.

y

x

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Measurement Range

• Measurement range (dynamic range or span):

the maximum & minimum values of the measurand that can be measured with a sensing system.

• All sensing systems are designed to perform

over a specified range.

• Signals outside of this range may be

unintelligible, cause unacceptably large inaccuracies, & may even result in irreversible damage to the sensor.

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Response and Recovery Time

• Response time: the time a sensing system requires to reach a stable value when it is exposed to a measurand.

• Recovery time: the time a sensing system

requires to reach a stable value when it is no longer exposed to a measurand.

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Contents

A. Introduction B. Sensors Characteristics I. Static Characteristics II. Dynamic Characteristics

C. Some Basic Measurement Methods D. Measurement Systems

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Dynamic Characteristics (1)

• Applied to time – varying measurands. • To describe the sensing system’s transient

properties.

• Can be used to define how accurately the

output signal is employed for the description of a time – varying measurand.

• Deal with issues such as the rate at which the output changes in response to a measurand alteration, and how these changes occur.

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Dynamic Characteristics (2)

( )y t

( )x t

Linear time invariant (LTI) system

n

1 

d

a

a

...  

n

n

a 1

a y t ( ) 0

1 

y t ( ) 1 n 

n d y t ( ) n dt

dt

( ) dy t dt

m

m

2

1 

d

d

...  

b m

b m

b 2

b x t ( ) 1

b 0

1 

x t ( ) 1 m 

x t ( ) 2 m 

dt

dt

dx t ( ) dt

( )x t

t

n

1 

d

a

a

...  

n

n

a 1

a y t ( ) 0

b 1

1 

y t ( ) 1 n 

n d y t ( ) n dt

dt

dy t ( ) dt

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Dynamic Characteristics (3)

( )y t

( )x t

Linear time invariant (LTI) system

or

y t ( )

K

a y t ( ) 0

b 1

b 1 a

0

Zero – order system

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Dynamic Characteristics (4)

( )y t

( )x t

Linear time invariant (LTI) system

(First – order system)

a 1

a y t ( ) 0

b 1

dy t ( ) dt

1

y t ( )

a dy t ( ) 1 dt a

b 1 a

0

0

0.8

y t ( )

K

,

1

0.6

K

e

)

0.6321

K

(1 1/ 

dy t ( ) dt

) t ( y

;

K

0.4

a 1 a

b 1 a

0

0

0.2

t  e 

)

y t ( )

K

(1

0

0

0.2

0.4

0.6

1

1.2

1.4

1.6

0.8 t (s)

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Dynamic Characteristics (5)

3.6km/h 1m/s

Ex. A car is equiped with altitude & temperature sensors & associated measurement systems. It is traveling up a hill at a constant vertical speed of 3.6 km/h. The temperature at the bottom of the hill is 20oC, Tx(x) = 20oC – 0.1x. The altitude sensing system is a zero – order response. The temperature sensing system has a first – order response with τ = 10s. a) What are the temperature & height measurements at 10, 20, 30, & 40s? b) What are the temperature measurements if τ = 1s? xv  x t ( )

1

t

t

  

v t x

x

y t ( )

x t ( )

10

T t ( ) m

T t ( ) x

dy t ( ) dt dT t ( ) m dt

( )mT t

xT t ( )

Linear time invariant (LTI) system

( )y t

( )x t

 

20 0.1x  20 0.1t 

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Dynamic Characteristics (6)

Ex. A car is equiped with altitude & temperature sensors & associated measurement systems. It is traveling up a hill at a constant vertical speed of 3.6 km/h. The temperature at the bottom of the hill is 20oC, Tx(x) = 20oC – 0.1x. The altitude sensing system is a zero – order response. The temperature sensing system has a first – order response with τ = 10s. a) What are the temperature & height measurements at 10, 20, 30, & 40s? b) What are the temperature measurements if τ = 1s?

t 2 0.01

 

10

20 0.1 t 

T t 0.1 ( ) m

T t ( ) m

dT t ( ) m dt

0.1

dt

t 0.1

t ( )

e

t 0.1

e

(2 0.01 )

0.1 t e dt

t 0.1 te dt C

t dt C 

 0.01

dT t ( ) m dt  e  

  2

T t ( ) m

t 0.1

t 0.1

e

t 0.1

t 0.1

t 0.1

t 0.1

20

e

0.1

te

0.1 t e dt

C

20

e

tde

C

e 

 0.1

t 0.1

t 0.1

e

e

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Dynamic Characteristics (7)

Ex. A car is equiped with altitude & temperature sensors & associated measurement systems. It is traveling up a hill at a constant vertical speed of 3.6 km/h. The temperature at the bottom of the hill is 20oC, Tx(x) = 20oC – 0.1x. The altitude sensing system is a zero – order response. The temperature sensing system has a first – order response with τ = 10s. a) What are the temperature & height measurements at 10, 20, 30, & 40s? b) What are the temperature measurements if τ = 1s?

t 0.1

t 0.1

t 0.1

t 0.1

t 0.1

20

e

0.1

te

0.1 t e dt

C

20

e

0.1

te

10

e

C

T t ( ) m

t 0.1

t 0.1

e

e

t 0.1

21 0.1 

t Ce 

0.1 0 

21 0.1 0

Ce

20

 

 

(0)

o 20 C

mT

1

C  

t 0.1

e

21 0.1 t 

mT t ( )

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Dynamic Characteristics (8)

Ex. A car is equiped with altitude & temperature sensors & associated measurement systems. It is traveling up a hill at a constant vertical speed of 3.6 km/h. The temperature at the bottom of the hill is 20oC, Tx(x) = 20oC – 0.1x. The altitude sensing system is a zero – order response. The temperature sensing system has a first – order response with τ = 10s. a) What are the temperature & height measurements at 10, 20, 30, & 40s? b) What are the temperature measurements if τ = 1s?

o

o

t 0.1

e

20 C 0.1 ; t 

21 C 0.1 t 

T x ( ) x

T t ( ) m

Time (s) Altitude (m) Real temp (oC) Measured temp (oC) Temp error (oC)

0 0 20 20 0

10 10 19 19.6321 0.6321

20 20 18 18.8647 0.8647

30 30 17 17.9502 0.9502

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40 40 16 16.9817 0.9817

Dynamic Characteristics (9)

o

o

t 0.1

e

20 C 0.1 ; t 

21 C 0.1 t 

Ex. T x ( ) x

T t ( ) m

20

)

Actual temperature Measured temperature

C o

19

18

( e r u t a r e p m e T

17

16

0

5

10

15

25

30

35

40

20 Time (s)/Altitude (m)

)

1

0.8

0.6

0.4

0.2

C (o r o r r e e r u t a r e p m e T

0

0

5

10

15

25

30

35

40

20 Time (s)/Altitude (m)

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Dynamic Characteristics (10)

Ex. A car is equiped with altitude & temperature sensors & associated measurement systems. It is traveling up a hill at a constant vertical speed of 3.6 km/h. The temperature at the bottom of the hill is 20oC, Tx(x) = 20oC – 0.1x. The altitude sensing system is a zero – order response. The temperature sensing system has a first – order response with τ = 10s. a) What are the temperature & height measurements at 10, 20, 30, & 40s? b) What are the temperature measurements if τ = 1s?

o

t 0.1

o

o

o

 

20 C 0.1 ; t  t 20 C 0.1 ; 

e t 21 C 0.1    0.1 t  e t 20.1 C 0.1  

10   1  

 

( ) T t m T t ( )  m

( ) T x x T x ( ) x Time (s) Altitude (m) Real temp (oC) Measured temp (oC)

Temp error (oC)

0 0 20 20 0

10 10 19 19.1 0.1

20 20 18 18.1 0.1

30 30 17 17.1 0.1

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40 40 16 16.1 0.1

Dynamic Characteristics (11)

 = 10

 = 1

20

20

)

)

Actual temperature Measured temperature

Actual temperature Measured temperature

C o

C o

19

19

18

18

( e r u t a r e p m e T

( e r u t a r e p m e T

17

17

16

16

0

5

10

15

25

30

35

40

0

5

10

15

25

30

35

40

20 Time (s)/Altitude (m)

20 Time (s)/Altitude (m)

)

)

1

1

0.8

0.8

C (o r o r r e

C (o r o r r e

0.6

0.6

e r u

e r u

t

t

0.4

0.4

0.2

0.2

a r e p m e T

a r e p m e T

0

0

0

5

10

15

25

30

35

40

0

5

10

15

25

30

35

40

20 Time (s)/Altitude (m)

20 Time (s)/Altitude (m)

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Dynamic Characteristics (12)

( )y t

( )x t

Linear time invariant (LTI) system

a

(Second – order system)

2

a 1

a y t ( ) 0

b 1

2 d y t ( ) 2 dt

( ) dy t dt

y t ( )

K

2 d y t ( ) 2 dt

dy t ( ) dt

1 2 

2  

0

:

the undamped natural frequency

2  

a a

2

the dampening ratio

 

a a 0 2 :

a 1 2

K

b 1 a

0

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y t ( )

K

2 d y t ( ) 2 dt

Dynamic Characteristics (13) 2 dy t ( )  dt 

1 2 

2

1.8

1.6

1.4

1.2

K

1

/ ) t ( y

0.8

0.6

0.4

0.2

 = 0  = 0.1  = 0.2  = 0.4  = 1.0  = 2.0  = 0.707

0

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1 Time (s)

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