Static/Dynamic Characteristics of Measurement Systems, Statistical Analysis and Curve Fitting

1.INTRODUCTION

An instrument is a device for determining the value or magnitude of a quantity or variable. As technology expands the demand for more accurate instruments increase and products new developments in instrument design and application. On the other hand, measurement is a process by which one can convert physical parameters to a meaningful number.

2.METHODS OF MEASUREMENTS

The methods of measurements may be classified according to the following types:

2.1. Direct Methods: In the direct method of measurement, we compare the unknown quantity directly with the primary or secondary standard. For example, if we want to measure the length of the bar, we will measure it with the help of the standard measuring tape or scale that acts as the secondary standard. Here we compare the unknown quantity directly with the standard scale. The scale is expressed as a numerical number and a unit.

The direct comparison method of measurement is not always accurate. In the above example of measuring the length, there is limited accuracy with which our eye can read the readings, which can be about 0.01 inch. Here the error does not occur because of the error in the standards, but because of the human limitations in taking the readings.

2.2. Indirect Methods: There is a number of quantities that cannot be measured directly by using some instrument. For instance, we cannot measure the strain in the bar due to the applied force directly. In such cases, indirect methods of measurements are used. In this method, the unknown quantity to be measured is converted into some other measurable quantity. Then we measure the measurable quantity. For example, the strain can be measured in terms of the electrical resistance of the bar.

3.CLASSIFICATION OF INSTRUMENTS

The instruments may be classified according to the following types:

3.1. Electrical and Electronic Instruments. The measuring instrument that uses the mechanical movement of electromagnetic meter to measure voltage, current, power, etc. is called an electrical measuring instrument. These instruments use the d'Arsonval meter. While any measurement system that uses a d'Arsonval meter with amplifiers to increase the sensitivity of measurements is called an electronic instrument.

3.2. Analogue and Digital Instruments. An analogue instrument is an instrument that uses an analogue signal to display the magnitude of quantity under measurement. The digital instrument uses a digital signal to indicate the results of measurement in digital form.

3.3. Absolute and Secondary Instruments. In an absolute instrument, the measured value is given in term of instrument constants and the deflection of one part of the instrument e.g. tangent galvanometer. In these instruments no calibrated scale is necessary. While in secondary instruments, the quantity of the measured values is obtained by observing the output indicated by these instruments.

4.CLASSIFICATION OF SECONDARY INSTRUMENTS

The secondary instrument may be classified into the following categories:

4.1. Indicating Instruments.

The magnitude of quantity being measured is obtained by deflection of the pointer on the scale, and the output is indicated either in analogue or digital form like ammeter, voltmeter, and wattmeter. Three forces were acting on the pointer to deflect it is proportional to the quantity being measured, these forces are of the following types:

(a) Deflecting Force. This force gives the pointer the initial force to move it from zero position, it's also called deflecting force.

(b) Controlling Force. This force control and limits the deflection of the pointer on scale which must be proportional to the measured value, and also ensure that the deflection is always the same for the same values.

This force is necessary to bring the pointer quickly to the measured value, and then stop without any oscillation.

4.2. Recording Instruments.

An instrument that makes a record in any recorded medium of the quantity being measured in order to save the information and use it at another time. The instruments like recording devices, X-Y plotter, and oscilloscope and recording instruments.

5.CHARACTERISTICS OF MEASUREMENT SYSTEMS

The characteristics of measurement systems are classified into the following two types:

Static Characteristics
Dynamic Characteristics

Both characteristics of measurements systems are discussed one by one in the following pages.

5.1. Static Characteristics

The static characteristic of a measurement instrument is the characteristics of the system when the input is either held constant or varying very slowly. The static characteristics are of the following types:

5.1.1 Sensitivity. The sensitivity of measurement is a measure of the change in instrument output that occurs when the quantity being measured changes by a given amount

5.1.2. Linearity. It is normally desired that the output reading of the instrument is linearly proportional to the quantity being measured. An instrument is considered linear if the relationship between output and input can be fitted in a line if it is not a straight line it should not be concluded that the instrument is inaccurate, it is a misconception.

5.1.3. Reproducibility. In the measurement, the given value may be repeated or measured assuming that environmental conditions are the same for each measurement_ We say that the measuring instruments have a certain amount of inherent uncertainty in their ability to reproduce the same output reading after some time.

5.1.4. Range and Span. It defines the maximum and minimum values of the inputs or the outputs for which the instrument is recommended to use. For example, for a temperature measuring instrument the input range may be 100-500°C and the output range maybe 4-20 mA. Span is the algebraic difference of the upper and lower limits of the range.

5.1.5. Static Error. This error shows the deviation of the true value from the desired value.

5.1.6. Loading Effects. It's the change of circuit parameter, characteristic, or behaviour due to instrument operation.

5.1.7. Accuracy and Precision

Accuracy is a closeness with which the instrument reading approaches the true value of the variable under measurement. Accuracy is the degree to which instrument reading match the true or accepted values. It indicates the ability of an instrument to indicate the true value of the quantity.

Accuracy refers to how closely the measured value of a quantity corresponds to its "true" value.

Precision is a measure of the reproducibility of the measurement i.e., its measure of the degree to which successive measurements differ from one other. It is the degree of agreement within a group of measurements or instruments. For example, if any resistance has a true value of 3.385,695 Ω, it always read 3.4 MΩ in scale reading.

Let us consider two voltmeters of the same model, both meters have knife-edged pointers and mirror-backed scales to avoid parallax, and they have calibrated scales. They may therefore be read to the same precision. If the value of the resistance in one meter changes considerably, its reading may be in error by a fairly large amount. Therefore the accuracy of the two meters may be quite different.

The precision is composed of two characteristics

Conformity
Significant Figures

Both conformity and significant figures are discussed one by one in the following pages.

Conformity

Consider, for example, that a resistor, whose true resistance is 3,385,695 Ω is measured by an ohmmeter. This consistently and repeatedly indicates 3.4 M Ω. The observer cannot read the true value from the scale. He estimates from the scale reading consistently a value of 3.4 MΩ. This is as close to the true value as he can read the scale by estimation. Although there are no deviations from the observed value, the error created by the lamination of the scale reading is a precision error. Conformity is necessary for measurements.

Significant Figures

An indication of the precision of the measurement is obtained from the number of significant figures in which the result is expressed. Significant figures convey actual information regarding the magnitude and the measurement precision of a quantity.

For example, if a resistor is specified as having a resistance of 65 Ω, its resistance value should be closer to 65 Ω than to 64 Ω or 66 Ω. If the value of resistor is described as 65.0 Ω., it means that its resistance is close to 65.0 Ω than it is to 64.9 Ω or 65.1 Ω. In 65 Ω there are two significant figures 6 and 5, while in 65.0 Ω there are three significant figures 6, 5 and 0.

5.1.8. Resolution

Resolution is the smallest amount of input signal change that the instrument can detect reliably. If the input is slowly increased from some arbitrary input value, it will again be found that output does not change at all until a certain increment is exceeded. This increment is called resolution or discrimination of the instrument. Thus the smallest increment in input that can be detected with certainty by an instrument is its resolution or discrimination.

5.2. Dynamic Characteristics

The dynamic characteristics of a measurement instrument describe the behaviour of the instrument when the desired input is not constant but varies rapidly with time. Following are the main types of dynamic characteristics:

5.2.1. Speed of Response.

It is defined as a rapidity with which a measurement system responds to a change in measured quantity. It gives information about how fast the system reacts to the changes in the input.

5.2.2. Measuring lag.

Every instrument takes some time to respond to the change in the measured variable. This retardation or delay in the response of the instrument is called measuring lag. The measuring lag is of the following two types:

(a) Retardation Lag. The response of the measurement system begins immediately after a change in measured quantity has occurred.

(b) Time Delay Lag. The measurement lags of this type are very small and are of the order of a fraction of a second and hence can be ignored. In this case, the response begins after the application of input and is called after "dead time". Such a delay shifts the response along the time axis and hence causes the dynamic error.

The largest change of input quantity for which there is no change in the measured quantity is known as a dead zone.

5.2.3. Fidelity. It is the ability of an instrument to produce a wave shape identical to the wave shape of input with respect to time. It also shows the change in quantity without dynamic error.

5.2.4. Dynamic Error. It is the difference between the true value changing with time & value indicated by measuring system without static errors

6.MEASUREMENT ERROR

No measurement can be made with perfection and accuracy, but it is important to find out what the accuracy actually is and how different errors have entered into the measurement. The error occurs due to several sources like human carelessness in taking a reading, calculating and in using instrument etc. Some of the time error is due to the instrument and environment effects.

Errors come from different sources and are classified in three types:

Gross Error
Systematic Errors
Random errors

6.1. Gross Error

The gross ‘nor occurs due to the human mistakes in reading or using the instruments. These errors cover human mistakes like in reading, calculating and recordings etc. It sometimes occurs due to incorrect adjustments of instruments.

The complete elimination of gross errors is impossible, but we can minimize them by the following ways:

It can be avoided by taking care while reading and recording the measurement data.
Taking more than one reading of the same quantity. At least three or more reading must be taken by different persons.

6.2. Systematic Errors

A systematic error is divided into three different categories: instrumental errors, environmental errors and observational errors.

6.2.1. Instrumental Errors

The instrument error generates due to the instrument itself. It is due to the inherent shortcomings in the instruments, misuse of the instruments, loading effects of instruments. For example in the D'Arsonval movement friction in bearings of various moving components may cause incorrect readings. There are so many kinds of instrument errors, depending on the type of instrument used.

Instrumental errors may be avoided by

(a) Selecting a suitable instrument for the particular measurement application

(b) Applying correction factors after determining the amount of instrumental error

6.2.2. Environmental Errors

Environmental errors arise as a result of environmental effects on instrument. It includes conditions in the area surrounding the instrument, such as the effects of changes in temperature, humidity, barometric pressure or of magnetic or electrostatic fields. For example, when making measurements with a steel rule, the temperature when the measurement is made might not be the same as that for which the rule was calibrated. Environmental errors may be avoided by

(a) Using the proper correction factor and information supplied by the manufacturer of the instrument.

(b) Using the arrangement which will keep the surrounding condition constant like the use of air condition, temperature-controlled enclosures etc.

6.2.3. Observational Errors

These errors occur due to the carelessness of operators while taking the reading. There are many sources of observational errors such as parallax error while reading a meter, wrong scale selection, the habits of individual observers etc.

To eliminate such observational errors, one should use instruments with mirrors, knife-edged pointers, etc. Now a day's digital display instruments are available, which are much more versatile.

6.3. Random Errors

These errors are due to unknown causes and occur even when all systematic errors have been accounted for. In some experiments, some random errors usually occur, but they become important in high accuracy work. These errors are due to friction in instrument movement, parallax errors between pointer and scale, mechanical vibrations, hysteresis in elastic members etc. When we measure a volume or weight, we observe reading on a scale of some kind. Scales by their very nature are limited to fixed increments of value, indicated by the division marks. The actual quantities we arc measuring, in contrast, can vary continuously. So there is an inherent limitation in how finally we can discriminate between two values that fall between the marked divisions of the measuring scale. The same problem remains if we substitute an instrument with a digital display. There will always be some point at which some value that lies between the smallest divisions must arbitrarily toggle between two numbers on the readout display. This introduces an element of randomness into the value we observe, even if the true value remains unchanged. These errors are of variable magnitude and sign and do not obey any known law. The presences of random errors become evident when different results are obtained on repeated measurements of one and the same quantity.

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