Download Uncertainties of measurement

Uncertainties of measurement
Laboratory task
A temperature sensor is connected as a voltage divider according to
the schematic diagram on Fig. 1. The temperature sensor is a
thermistor type B57164K [2] with nominal resistance for 25 °C R2
= 2200 Ω ± 10 %. The thermistor resistance dependence on
temperature is non linear and is given by characteristic No. 1013 in
Appendix.
The connection is powered by a 9V regulated switching power
supply type RQT666K [3].
The measured output voltage U1 (variable with temperature) is
measured with a voltmeter type MT-1232 [5] on range 400 mV DC.
Used resistor (nominal value: R1 = 270 kΩ) is a precision metal
film resistor with accuracy ± 0.1%.
Task:
a) Calculate uncertainty of measurement type A for voltage U1.
b) Calculate type B uncertainty of temperature measurement uT by
considering the following uncertainty sources (for these tasks
consider the temperature in the lab to be exactly 25 °C):
Fig.
1 - Schematic
diagram of temperature
sensor
- Uncertainty caused by power supply voltage U0 variations = uB1
- Uncertainty of R2 calculation caused by uncertainty of power
supply voltage U0, uncertainty of measurement of voltage U1 and
R1 tolerance = uB2
Write results as Temperature ± uncertainty and state the used probability distribution function
and used coverage factor.
c) In conclusions write what other sources of uncertainty could be added to the calculations
for a more precise result
Solution
Important note: Remember, you have to really repeat the whole experiment, it means to
turn off and on the power supply. Use the switch on the fixture for this. It is not sufficient
to just read 9 following values from the voltmeter.
a) Uncertainty type A
Uncertainty type A is obtained by repeating the experiment and by statistical evaluation.
Tab. 3 - measured and calculated values
i
U1
(mV)
1
2
3
4
5
6
7
8
Sample mean
n
x=
∑x
i =1
i
n
=
(1)
Estimated standard deviation of the mean = uncertainty type A
∑ ( x − x)
n
u A = sx =
sx
n
=
i =1
2
i
(2)
n ( n − 1)
As the experiment was repeated only 8 times, the interval for uncertainty type A has to
be extended => corrected uncertainty type A needs to be calculated. In other words if the
experiment is repeated only 3 times the result is much less sure than it is if the experiment is
repeated 100 times.
The values of the correction coefficient are given by the coverage factor from the Student’s tdistribution table. As at the end we will write the result in form: average ± uncertainty, we
take only ½ of the value in Student’s t-distribution table. Those values are shown in Tab. 4
directly. If the result would be written as: average; uncertainty, then we would take the whole
interval.
Tab. 1 – Coverage factor for normal distribution of uncertainty type A for p= 95,45 % (rounded)
i
100
(number
of
measurements)
k
1,0
(coverage
factor)
25
10
9
8
7
6
5
4
3
2
1,1
1,2
1,2
1,2 1,3 1,3 1,4 1,7 2,3 7,0
Corrected uncertainty type A is
u AK = k ⋅ u A =
(3)
b) Uncertainty type B
- Uncertainty uU0 in power supply voltage
From manufacturer specifications on Fig. 3 it can see that the output voltage should be
9,15 V ± 0,15 V when the power supply is unloaded. This assumption can be made as the
circuit current is very small; it was measured to be approximately 120 µA.
As the manufacturer gives us an interval and not uncertainty, we have to calculate it. The
interval 9,15 V ± 0,15 V is where the voltage is with certainty “almost 100 %”.
If interval ∆ of the used value is known then standard uncertainty is found from
∆
uU 0 =
k
(4)
where ∆ is the semi-range (or half-width) between the upper and lower limits and k is a
divisor dependent on the shape of the probability distribution function of our variable
(rectangular, normal, U-Shape etc.). Its values are shown in Tab. 5
Tab. 5 - divisors for various probability distribution functions
probability
distribution
function
Rectangular
(for
probability
57.74 %)
Coverage
factor k
3
normal
U(Gaussian) Shape
for
2σ
(95,45 %)
2
2
triangular
6
If we consider a normal (Gaussian) probability distribution of power supply output voltage,
the uncertainty caused by power supply voltage variations will be
uU 0 =
∆
=
k
(5)
- Calculation of uncertainty uR2 caused by uncertainty uncertainty uU0 in power supply
voltage, uncertainty of measurement of voltage U1 and R1 tolerance = uB2
To calculate uncertainty of measurement of R2 we have to first analyze the circuit. The
connection is a voltage divider composed of resistor R1 and temperature sensor R2.
Output voltage of this divider is
U 1 = f (U 0, R1, R 2)
(6)
From circuit analysis it is known
R2
U1
U1 =
⋅U 0 ⇒ R 2 =
R1
R 2 + R1
U 0 − U1
(7)
(8)
Uncertainty uR2 will be given by uncertainties of all variables in equation(8), i.e. by
uncertainty of U0, U1 and R1. The individual components are calculated as follows
- Uncertainty uU0 in power supply voltage
Uncertainty caused by power supply voltage variations uB1 was calculated earlier as
uU 0 =
- Uncertainty caused by resistance R1 tolerance uR1
To calculate uncertainty of resistance R1, uR1, available manufacturer data is used,
R1 = 270 kΩ ± 0,1 % => ∆ R1 = R1 ⋅ 0,1% / 100 = ± 0,27 kΩ . If we assume a normal
(Gaussian) distribution of resistance for 2σ then
u R1 =
∆ R1
=
k
(9)
- Uncertainty of measurement of voltage U1 uU1
From voltmeter manufacturer specifications on Fig. 5 and Fig. 6 it is found that on range 400
mV, accuracy is ±(0,5 % of reading + 4 digits), resolution 100 µV.
Considering the worst case (highest inaccuracy) on this range (maximal voltage), we get
± (0,5 % of reading + 4 digits) = ± ( 0,5 % x 400 mV + 4x100 µV) = ± 2,4 mV
Uncertainty for a rectangular distribution is
uU 1 =
accuracy
=
2 3
(10)
- Uncertainty of R2 calculation = uR2
In a general case, to calculate the uncertainty of a variable given as a function f of input
variables x1, x2, ..., xn it is necessary to calculate a square root of sum of squares of partial
derivatives from function f by all variables x1, x2, ..., xn multiplied by uncertainty of nominal
values of variables u x1 N , u x2 N ,..., u xnN
This is called law of propagation of uncertainty
2
2
 ∂f
  ∂f

 ∂f

u B = 
⋅ u x1 N  + 
⋅ u x2 N  + K + 
⋅ u xnN 
 ∂x1
  ∂x2

 ∂xn

2
(11)
Note 1.: Uncertainties of measurement always add together. If one uncertainty is much larger
and others are small, the large one will be dominant and no improvement in the smaller
uncertainties in the whole measuring chain will improve much the combined uncertainty. It
will be the weakest link in the whole chain. In other words, the most significant part of total
uncertainty will be caused by the largest uncertainty.
As not correlated variables are assumed, in our case R 2 = f (U 0,U 1, R1) , the uncertainty is
 ∂R 2
  ∂R 2
  ∂R 2

uR 2 = 
⋅ uU 0  + 
⋅ uU 1  + 
⋅ u R1 
 ∂U 0
  ∂U 1
  ∂R1

2
R2 =
2
U1
R1
U 0 − U1
2
(12)
(13)
− ( R1 ⋅ U 1)
∂R 2
⋅ uU 0 =
⋅ uU 0 =
2
∂U 0
(U 0 − U 1)
(14)
 R1 ⋅ U 1
∂R 2
R1 
⋅ uU 1 = 
+
 ⋅ uU 1 =
2
∂U 1
 (U 0 − U 1) U 0 − U 1 
(15)
∂R 2
U1
⋅ u R1 =
⋅ u R1 =
∂R1
U 0 − U1
(16)
uR 2 =
(17)
The uncertainty uR2 shows the uncertainty of R2 measurement based on known properties of
the circuit. To recalculate this to uncertainty of temperature measurement, the dependence of
thermistor resistance as a function of temperature has to be known. From the manufacturer
data on Fig. 2 it was determined, that in the range 10 °C to 50 °C the dependence can be
approximated with an exponential function
R2 [Ω]
ln
6295
R 2 [ Ω ] = 6295 ⋅ e −0,0414⋅T [°C ] ⇒ T [ °C ] =
(18)
−0, 0414
- Uncertainty caused by resistance R2 tolerance uR2TOL
Equation (18) gives us an approximated dependency of resistance on temperature. The
manufacture values are given with a tolerance ± 10 %. This tolerance of the nominal value has
also to be considered for temperature uncertainty calculations.
R2 = 2220 Ω ± 10 % => ∆ R 2 = R 2 ⋅10% /100 = ±220 Ω . If we assume a normal (Gaussian)
distribution of resistance for 2σ then uncertainty is
u R 2TOL =
∆R2
=
k
(19)
- Uncertainty of temperature measurement uT
The uncertainty of temperature T measurement is given by uncertainty caused by the
properties of the connection given by uncertainty uR2 and by uncertainty caused by thermistor
R2 tolerance uR2TOL
 ∂T
  ∂T

uT = 
⋅ uR 2  + 
⋅ u R 2TOL 
 ∂R 2
  ∂R 2

2
2
(20)
∂T
−5000
=
∂R 2 207 ⋅ R 2
∂T
−5000
⋅ uR 2 =
⋅ uR 2 =
∂R 2
207 ⋅ R 2
∂T
−5000
⋅ u R 2TOL =
⋅ uR 2TOL =
∂R 2
207 ⋅ R 2
uT =
(21)
(22)
(23)
(24)
Write result as:
Temperature ± uncertainty.
Uncertainty has been evaluated assuming normal distribution and coverage factor k = 2
(probability 95,45 %)
Appendix
Fig. 2 - Thermistor resistance - temperature dependence
Fig. 3 - Power supply specifications
Fig. 4 - Accuracy specifications
Fig. 5 - MT-1232 multimeter accuracy for DC voltage
References
[1] Bell S.: A Beginner's Guide to Uncertainty of Measurement, online (19.12.2010) on
http://www.wmo.int/pages/prog/gcos/documents/gruanmanuals/UK_NPL/mgpg11.pdf
[2] NTC thermistors for temperature measurement, online (26.1.2011) on
http://www.epcos.com/inf/50/db/ntc_09/LeadedDisks__B57164__K164.pdf
[3]
Power
supply
specifications,
online
(26.1.2011)
on
http://www.gme.cz/_dokumentace/dokumenty/751/751-518/dsh.751-518.1.pdf
[5]MT-1232 online (26.1.2011) on http://www.prokits.com.tw/pkjpg/pic7/MT-1232.pdf
[6] Taylor B. N., Kuyatt Ch. E.: Guidelines for Evaluating and Expressing the Uncertainty of
NIST
Measurement
Results,
online
(26.1.2011)
on
http://www.nist.gov/pml/pubs/tn1297/index.cfm
[7] Appendix V. Uncertainties and Error Propagation, online (26.1.2011) on
http://physicslabs.phys.cwru.edu/MECH/Manual/Appendix_V_Error%20Prop.pdf