17 Measurements I

Vinay Gupta

    Measurements

To measure a given quantity, a comparison is made between the quantity and a well-defined standard. The result of measurement is expressed in numbers as it is a mere comparison. For the results to be meaningful, the predefined standard used for comparison must be accurately defined and commonly accepted. Secondly, the instruments used for measurement must be provable.

The progress of Science and Technology depends majorly upon the progress in measurement techniques. The development of new phenomena and relationships in Science and Technology make the measurements to be further more significant. These new advancements are of utility only if they are backed by actual measurements which approve the authority of a hypothesis. The measurements also improve the understanding of the proposed work.

Methods of Measurement

Generally the methods of measurement are organized in two categories:

1. Direct Methods: These involve direct comparison of the quantity to be measured with the pre-defined standard. The results in direct measurement are given in numerical numbers with a well-defined unit. These are commonly used for the measurement of physical quantities such as length, mass and time. Say, we plan to measure the length of a bar in units of meter. We can measure the length with a precision of say 0.45mm. Thus, due to human factors, it may not be possible to perform very accurate measurements. Thus the direct measurements may not be used with high degree of accuracy due to human errors.
2. Indirect Methods: Direct methods are not very feasible and are inaccurate due to the involvement of human factors. Also, the sensitivity is compromised in the direct methods of measurement. Thus, direct methods are rarely used and are generally not preferred. In today’s applications, Measurement Systems are used which are called as indirect methods of measurement. These consist of transducing elements which convert the quantity to be measured in analogous form. The signal is then processed by an intermediate means which is then fed to the end device and the results of measurement are presented.

    Errors in Measurements and their Statistical Analysis

Measurement involves comparing the unknown quantity with a pre-defined accepted standard quantity. A measuring instrument is required to perform this comparison for the measurement. It is understood that measurement of any quantity with full accuracy is not possible and thus a study of errors is of utmost importance for the study of measuring processes. Measurements so performed cannot be classified as good or bad based on their degree of accuracy but it depends upon the adequacy of the measurements in the provided conditions. In order to understand the accuracy of a measurement, we need to study the different types of errors which may arise in the measurements and how can we deal with them.

1. Absolute Error: True value of quantity is the value which is observed by making measurements with standards and measuring instruments. However, there is always a difference in between the measured and true or exact value of a given unknown quantity. The difference in between the measured value Vm and the true value V of the given quantity is called as the absolute error of the measurement, δV i.e. δV = Vm – V
2. Relative Error or Percentage Error: The precise accuracy of a measurement cannot be evaluated by the absolute error. For example, an absolute error of 1cm is infinitesimal in determining the circumference of a cricket ground but same absolute error of 1cm is not admissible for a meter scale. Relative error is the ratio of absolute error to the true value of the quantity to be measured.

     i.e.   εr = δV / V

The relative error may be quoted as a fraction or may be expressed in percentage as well i.e. Percentage error = εr * 100

For example: if R = 100 ohm and δR = ± 10 ohm

Thus, relative error εr = δR / R = ± 10/100 = ± 0.1

Percentage limiting error % εr = 0.1 * 100 = ± 10 %.

3. Guarantee Error: The design, material and workmanship of an instrument define its accuracy and precision. Which instrument has to be used for a particular application depends mainly upon the accuracy desired in the measurement and that will ultimately decide the cost to be invested in the purchase of a given instrument. Mostly, the accuracy of an instrument is guaranteed to be within a defined percentage of its full scale reading. Thus, manufacturers of instruments specify the deviations from a nominal value of a given quantity. The limits of these deviations from the true value are called as Guarantee Errors.

It can be said that the manufacturer guarantees or claims that the error in the instrument he is selling is not more than the limit set. The value of a given quantity having a nominal value Vs and a maximum error or limiting error of ±δV must have a magnitude Va between the limits Vs-δV and Vs+δV or Actual value Va = Vs±δV.

for example, the nominal magnitude of a resistor is 100 ohm with a ‘limiting error of ±5ohm

The magnitude of the resistor will be between the limits:

R = 100 ±5 ohm or R ≥ 95 ohm and R ≤ 105 ohm

In other words the seller guarantees that the value of resistance of the resistor lies between 95 ohm and 105 ohm.

4. Systematic Errors: These kind of errors can be further classified in the following three types:

a. Instrumental Error

b. Environmental Error

c. Observational Error

(a) Instrumental Error: These kinds of errors arise due to several reasons such as i) inherent shortcoming of the instrument ii) due to misuse of the instruments and iii) loading effects.

Inherent errors refer to the inherent shortcomings of the instruments and are related to their mechanical structure. Reason for this kind of error may be due to anything from construction or operation of the measuring device to calibration of the instrument. In order to make precise measurements, one must be able to diagnose these errors as it is highly feasible to decrease them to a great extent. These errors can be easily reduced by carefully planning the procedure of measurements. One can introduce correction factors in the measurement after ascertaining the instrumental errors. Re-calibration of the instrument can also be performed to reduce the inherent errors.

Inherent errors in the measurements may also arise due to the mistake of the system operator such as forgetting to adjust zero of the machine, wrong initial adjustments etc. Such errors are however, not fatal to the instrument, but lead to erroneous results and sometimes lead to permanent damage to the machine. Loading effects may also arise in the measurements mainly due to improper use of the instruments. For example, a well calibrated voltmeter in good working condition may give wrong value when connected to a very high resistance circuit, while the same voltmeter would provide correct values when connected to a lower resistance circuit. This explains that the voltmeter has a loading effect on the circuit.

(b) Environmental Errors: These errors are related to the conditions of the surrounding area of the measuring device. These external effects include temperature, pressure, vibrations, dust, humidity or the external magnetic or electrostatic fields. Several corrective measures can be employed to reduce these errors. One may try to keep the ambient conditions as stable as possible such as by keeping the instrument in a temperature controlled closure. Resistor materials should be chosen in such a way that they may have very low temperature coefficient. By hermetically sealing the instruments, one can reduce the effect of humidity completely.

(c) Observational Errors: These are mainly related human errors. For example, Parallax errors arise in voltmeters where the pointer rests slightly above the scale. To avoid this error highly precise voltmeters are present with mirrored scales. Also, voltmeters can be used where the scale and pointer are in the same plane. Also, modern electrical instruments are provided with digital displays and hence the observational errors are completely eliminated.

5. Random Errors: It is always observed that the experiments show difference from one reading to other even when we have already removed all the possible errors from the experimental setup. These kinds of errors occur due to a variety of factors which change or fluctuate from one measurement to another and are just due to chance. Our measurements are influenced by several known and unknown factors. All the disturbances about which we are unaware can be classified together as Residual or Random. As these errors stay intact even after removing systematic errors, these are called as Residual errors. These errors are variable in nature and do not follow any defined rule.

6. Central Value: The random errors are caused by several small and variable factors. These variables may be either additive or subtractive in nature for the given value of quantity to be measured. If the positive and negative effects in the several measurements, so taken, are nearly small then, the resultant errors are small. If we have taken a large number of samples and the plus effects are equal to the negative effects, they shall cancel each other and we would obtain the scatter around a Central Value. The effect of random errors can be minimized by taking several measurements of the given quantity under same conditions and taking their arithmetic mean. This mean-value can be considered as the most probable value of the measured quantity. If the errors are truly random, a plot of the readings gives a typical error curve.

Statistical Treatment of Data

The experimental data is generally obtained in two forms of tests :

(i)   Multisample test and

(ii)  Single-sample test

Multisample Test: Repeated measurement of a given quantity are performed using different test conditions such as employing different instruments, different ways of measurement and by employing various observers. Multisample test doesn’t mean to make measurements with the same procedure technique, equipment and same observer.

Single Sample Test: If a single measurement (or succession of measurements) is performed under exactly same conditions except for time, then it is called as single-sample test.

Questionnaire

1. Random errors can be corrected by following proper experimental procedures. (True/False)
2. Environmental errors can be reduced by providing proper ambient conditions such as maintaining given temperature and humidity conditions. (True/False)
3. Relative error is defined as the ratio of ______________.
4. The two broad categories of methods of measurement are __________________.
5. One of the kind of systematic errors is:

(a) Random error

(b) Environmental error

(c) Guarantee error

(d) Relative error

6. The length of a meter stick as measured by an observer using a cm scale is observed to be 99.8cm. The absolute error in the measurement is:

(a) 02cm

(b) 01cm

(c) 1cm

(d) 2cm

References:

1. Electrical Measurements and Instrumentation by U. A. Bakshi