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Thursday, December 2, 2021

Errors during the measurement process


3.6.3 Total error when combining multiple measurements

The final case to be covered is where the final measurement is calculated from several measurements that are combined together in a way that involves more than one type of arithmetic operation. For example, the density of a rectangular-sided solid block of material can be calculated from measurements of its mass divided by the product of measurements of its length, height and width. The errors involved in each stage of arithmetic are cumulative, and so the total measurement error can be calculated by adding together the two error values associated with the two multiplication stages involved in calculating the volume and then calculating the error in the final arithmetic operation when the mass is divided by the volume.

Example 3.10

A rectangular-sided block has edges of lengths a, b and c, and its mass is m. If the values and possible errors in quantities a, b, c and m are as shown below, calculate the value of density and the possible error in this value.

                         a = 100 mm ± 1%, b = 200 mm ± 1%, c = 300 mm ± 1%, m = 20 kg ± 0.5%.

Solution

Value of ab = 0.02 m2 ± 2% (possible error = 1% + 1% = 2%)

Value of (ab)c = 0.006 m3 ± 3% (possible error = 2% + 1% = 3%)

Value of m/abc = 20/0.006 = 3330 kg/m3 ± 3.5% (possible error = 3% + 0.5% = 3.5%)

3.7 Self-test questions

3.1 Explain the difference between systematic and random errors. What are the typical sources of these two types of error?

3.3 Suppose that the components in the circuit shown in Figure 3.1(a) have the following values:

                          R1 = 330 Ω ; R2 = 1000 Ω ; R3 = 1200 Ω ; R4 = 220 Ω ; R5 = 270 Ω .

If the instrument measuring the output voltage across AB has a resistance of 5000 Ω , what is the measurement error caused by the loading effect of this instrument?

3.4 Instruments are normally calibrated and their characteristics defined for partic[1]ular standard ambient conditions. What procedures are normally taken to avoid measurement errors when using instruments that are subjected to changing ambient conditions?

3.5 The voltage across a resistance R5 in the circuit of Figure 3.10 is to be measured by a voltmeter connected across it.

(a) If the voltmeter has an internal resistance (Rm) of 4750 Ω , what is the measure[1]ment error?

(b) What value would the voltmeter internal resistance need to be in order to reduce the measurement error to 1%?

3.6 In the circuit shown in Figure 3.11, the current flowing between A and B is measured by an ammeter whose internal resistance is 100 Ω . What is the measure[1]ment error caused by the resistance of the measuring instrument?

3.7 What steps can be taken to reduce the effect of environmental inputs in measurement systems?

3.8 The output of a potentiometer is measured by a voltmeter having a resistance Rm, as shown in Figure 3.12. Rt is the resistance of the total length Xt of the potentiometer and Ri is the resistance between the wiper and common point C for a general wiper position Xi. Show that the measurement error due to the resistance





Rm of the measuring instrument is given by:


Hence show that the maximum error occurs when Xi is approximately equal to 2Xt/3. (Hint – differentiate the error expression with respect to Ri and set to 0. Note that maximum error does not occur exactly at Xi = 2Xt/3, but this value is very close to the position where the maximum error occurs.)

3.9 In a survey of 15 owners of a certain model of car, the following figures for average petrol consumption were reported.

         25.5 30.3 31.1 29.6 32.4 39.4 28.9 30.0 33.3 31.4 29.5 30.5 31.7 33.0 29.2

Calculate the mean value, the median value and the standard deviation of the data set.

3.10 (a) What do you understand by the term probability density function?

(b) Write down an expression for a Gaussian probability density function of given mean value µ and standard deviation α and show how you would obtain the best estimate of these two quantities from a sample of population n.

(c) The following ten measurements are made of the output voltage from a highgain amplifier that is contaminated due to noise fluctuations:

                      1.53, 1.57, 1.54, 1.54, 1.50, 1.51, 1.55, 1.54, 1.56, 1.53

Determine the mean value and standard deviation. Hence estimate the accuracy to which the mean value is determined from these ten measurements. If one thousand measurements were taken, instead of ten, but α remained the same, by how much would the accuracy of the calculated mean value be improved?

3.11 The following measurements were taken with an analogue meter of the current flowing in a circuit (the circuit was in steady state and therefore, although the measurements varied due to random errors, the current flowing was actually constant):

21.5 mA, 22.1 mA, 21.3 mA, 21.7 mA, 22.0 mA, 22.2 mA, 21.8 mA, 21.4 mA, 21.9 mA, 22.1 mA

Calculate the mean value, the deviations from the mean and the standard deviation.

3.12 The measurements in a data set are subject to random errors but it is known that the data set fits a Gaussian distribution. Use standard Gaussian tables to determine the percentage of measurements that lie within the boundaries of ±1.5 α, where α is the standard deviation of the measurements.

3.13 The thickness of a set of gaskets varies because of random manufacturing distur[1]bances but the thickness values measured belong to a Gaussian distribution. If the mean thickness is 3 mm and the standard deviation is 0.25, calculate the percentage of gaskets that have a thickness greater than 2.5 mm.

3.14 A 3 volt d.c. power source required for a circuit is obtained by connecting together two 1.5 V batteries in series. If the error in the voltage output of each battery is specified as ±1%, calculate the likely maximum possible error in the 3 volt power source that they make up.

3.15 In order to calculate the heat loss through the wall of a building, it is necessary to know the temperature difference between the inside and outside walls. If temper[1]atures of 5°C and 20°C are measured on each side of the wall by mercury-in-glass thermometers with a range of 0°C to +50°C and a quoted inaccuracy figure of ±1% of full-scale reading, calculate the likely maximum possible error in the calculated figure for the temperature difference.

3.16 The power dissipated in a car headlight is calculated by measuring the d.c. voltage drop across it and the current flowing through it (P = V × I). If the possible errors in the measured voltage and current values are š1% and ±2% respectively, calculate the likely maximum possible error in the power value deduced.

3.17 The resistance of a carbon resistor is measured by applying a d.c. voltage across it and measuring the current flowing (R = V/I). If the voltage and current values are measured as 10 ± 0.1 V and 214 ± 5 mA respectively, express the value of the carbon resistor.

3.18 The density (d) of a liquid is calculated by measuring its depth (c) in a calibrated rectangular tank and then emptying it into a mass measuring system. The length and width of the tank are (a) and (b) respectively and thus the density is given by:

                                                                 d = m/(a × b × c)

where m is the measured mass of the liquid emptied out.

If the possible errors in the measurements of a, b, c and m are 1%, 1%, 2% and 0.5% respectively, determine the likely maximum possible error in the calculated value of the density (d).

3.19 The volume flow rate of a liquid is calculated by allowing the liquid to flow into a cylindrical tank (stood on its flat end) and measuring the height of the liquid surface before and after the liquid has flowed for 10 minutes. The volume collected after 10 minutes is given by:

Volume = (h2 - h1)π(d/2)2

where h1 and h2 are the starting and finishing surface heights and d is the measured diameter of the tank.

(a) If h1 = 2 m, h2 = 3 m and d = 2 m, calculate the volume flow rate in m3/min.

(b) If the possible error in each measurement h1, h2 and d is ±1%, determine the likely maximum possible error in the calculated value of volume flow rate.


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