The e.m.f. at a thermocouple junction is 645 µV at the steam point, 3375 uV at the zinc point and 9149 uV at the silver point. Given that the e.m.f.-temperature relationship is of the form E(T) = a,T+aT + a,T' (T in °C), find a,, a, and as.

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TIC CHARACTERISTICS OF MEASUREMENT SYSTEM ELEMENTS
Problems
2.1
The e.m.f. at a thermocouple junction is 645 µV at the steam point, 3375 uV at the zinc point
and 9149 µV at the silver point. Given that the e.m.f.-temperature relationship is of the form
E(T) = a,T+ a,T + a,T' (T in °C), find a,, a, and a;.
2.2 The resistance R(0) of a thermistor at temperature 0K is given by R(0) = a exp(Bie). Given
that the resistance at the ice point (0 = 273.15 K) is 9.00 k2 and the resistance at the steam
point is 0.50 k2, find the resistance at 25 °C.
2.3 A displacement sensor has an input range of 0.0 to 3.0 cm and a standard supply voltage
V = 0.5 volts. Using the calibration results given in the table, estimate:
(а)
The maximum non-linearity as a percentage of f.s.d.
(b)
The constants K, Ky associated with supply voltage variations.
(c)
The slope K of the ideal straight line.
Displacement x cm
Output voltage millivolts (V, = 0.5)
Output voltage millivolts (V, = 0.6)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
16.5
32.0
44.0
51.5
55.5
58.0
0.0
21.0
41.5
56.0
65.0
70,5
74.0
A liquid level sensor has an input range of 0 to 15 cm. Use the calibration results given in the
table to estimate the maximum hysteresis as a percentage of f.s.d.
2.4
Level h cm
Output volts h increasing 0.00 0.35 1.42 2.40 3.43 4.35 5.61
Output volts h decreasing 0.14 1.25 2.32 3.55 4.43 5.70 6.78
0.0
1.5
3.0
4.5 6.0
7.5
9.0
10.5
12.0
13.5
15.0
6.50
7.77 8.85 10.2
7.80
8.87
9.65 10.2
2.5 A repeatability test on a vortex flowmeter yielded the following 35 values of frequency
corresponding to a constant flow rate of 1.4 x 10 m' s: 208.6; 208.3; 208.7; 208.5; 208.8;
207.6; 208.9; 209.1; 208.2; 208.4; 208.1; 209.2: 209.6; 208.6; 208.5; 207.4; 210.2; 209.2; 208.7;
208.4; 207.7; 208.9; 208.7; 208.0; 209.0; 208.1; 209,3; 208.2; 208.6; 209.4; 207.6; 208.1; 208,8;
209.2; 209.7 Hz.
Using equal intervals of width 0.5 Hz, plot a histogram of probability density values,
(a)
(b) Calculate the mean and standard deviation of the data.
(c)
Sketch a normal probability density function with the mean and standard deviation
calculated in (b) on the histogram drawn in (a).
2.6
A platinum resistance sensor is used to interpolate between the triple point of water (0 °C),
the boiling point of water (100 °C) and the freezing point of zinc (419.6 °C). The corre-
sponding resistance values are 100.0 2, 138.5 2 and 253.7 2. The algebraic form of the inter-
polation equation is:
R, = R,(1 + aT+ BT)
R, 2 = resistance at T°C
R, 2 = resistance at 0 °C
a, B= constants.
where
Find the numerical form of the interpolation equation.
PROBLEMS 33
2.7
The following results were obtained when a pressure transducer was tested in a laboratory under
the following conditions:
Ambient temperature 20 °C, supply voltage 10 V (standard)
Ambient temperature 20 °C, supply voltage 12 V
Ambient temperature 25 °C, supply voltage 10 V
II
III
Input (barg)
Output (mA)
4
6
8.
10
I
4
7.2
10.4
13.6
16.8
20
II
4
8.4
12.8
17.2
21.6
28
III
6.
9.2
12.4
15.6
18.8
22
Determine the values of K, K, a and K associated with the generalised model equa-
tion O = (K + K,ylI + a + K,I,
(b)
(a)
Predict an output value when the input is 5 barg, V = 12 V and ambient temperature
is 25 °C.
Basic problems
Transcribed Image Text:TIC CHARACTERISTICS OF MEASUREMENT SYSTEM ELEMENTS Problems 2.1 The e.m.f. at a thermocouple junction is 645 µV at the steam point, 3375 uV at the zinc point and 9149 µV at the silver point. Given that the e.m.f.-temperature relationship is of the form E(T) = a,T+ a,T + a,T' (T in °C), find a,, a, and a;. 2.2 The resistance R(0) of a thermistor at temperature 0K is given by R(0) = a exp(Bie). Given that the resistance at the ice point (0 = 273.15 K) is 9.00 k2 and the resistance at the steam point is 0.50 k2, find the resistance at 25 °C. 2.3 A displacement sensor has an input range of 0.0 to 3.0 cm and a standard supply voltage V = 0.5 volts. Using the calibration results given in the table, estimate: (а) The maximum non-linearity as a percentage of f.s.d. (b) The constants K, Ky associated with supply voltage variations. (c) The slope K of the ideal straight line. Displacement x cm Output voltage millivolts (V, = 0.5) Output voltage millivolts (V, = 0.6) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 16.5 32.0 44.0 51.5 55.5 58.0 0.0 21.0 41.5 56.0 65.0 70,5 74.0 A liquid level sensor has an input range of 0 to 15 cm. Use the calibration results given in the table to estimate the maximum hysteresis as a percentage of f.s.d. 2.4 Level h cm Output volts h increasing 0.00 0.35 1.42 2.40 3.43 4.35 5.61 Output volts h decreasing 0.14 1.25 2.32 3.55 4.43 5.70 6.78 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 6.50 7.77 8.85 10.2 7.80 8.87 9.65 10.2 2.5 A repeatability test on a vortex flowmeter yielded the following 35 values of frequency corresponding to a constant flow rate of 1.4 x 10 m' s: 208.6; 208.3; 208.7; 208.5; 208.8; 207.6; 208.9; 209.1; 208.2; 208.4; 208.1; 209.2: 209.6; 208.6; 208.5; 207.4; 210.2; 209.2; 208.7; 208.4; 207.7; 208.9; 208.7; 208.0; 209.0; 208.1; 209,3; 208.2; 208.6; 209.4; 207.6; 208.1; 208,8; 209.2; 209.7 Hz. Using equal intervals of width 0.5 Hz, plot a histogram of probability density values, (a) (b) Calculate the mean and standard deviation of the data. (c) Sketch a normal probability density function with the mean and standard deviation calculated in (b) on the histogram drawn in (a). 2.6 A platinum resistance sensor is used to interpolate between the triple point of water (0 °C), the boiling point of water (100 °C) and the freezing point of zinc (419.6 °C). The corre- sponding resistance values are 100.0 2, 138.5 2 and 253.7 2. The algebraic form of the inter- polation equation is: R, = R,(1 + aT+ BT) R, 2 = resistance at T°C R, 2 = resistance at 0 °C a, B= constants. where Find the numerical form of the interpolation equation. PROBLEMS 33 2.7 The following results were obtained when a pressure transducer was tested in a laboratory under the following conditions: Ambient temperature 20 °C, supply voltage 10 V (standard) Ambient temperature 20 °C, supply voltage 12 V Ambient temperature 25 °C, supply voltage 10 V II III Input (barg) Output (mA) 4 6 8. 10 I 4 7.2 10.4 13.6 16.8 20 II 4 8.4 12.8 17.2 21.6 28 III 6. 9.2 12.4 15.6 18.8 22 Determine the values of K, K, a and K associated with the generalised model equa- tion O = (K + K,ylI + a + K,I, (b) (a) Predict an output value when the input is 5 barg, V = 12 V and ambient temperature is 25 °C. Basic problems
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