Linear calibration function

Linear calibration function

Linear calibration function//Quite often measuring instruments are received for calibration. Scale of the measuring instrument is calibrated at few points only. The correction or the value of standard input is assigned at those selected points of its scale and uncertainty of measurement is also stated at those points only. In most cases, the values of the standard input versus scale readings are given at the selected points. When an instrument is used in the field, the scale readings are recorded, which in general, may not be the same points at which the instrument was calibrated. The correct value is obtained from the corrections at the two nearest calibrated points just by linear manipulation. In this method only small interval containing the observed reading is considered which may not be always justified. However, it is advisable to consider all the points at which calibration is carried out. It is, therefore, necessary that a mathematical relation between the scale reading and standard input is given. So that the user can substitute the value of observed scale reading in the relation and get the value of the input to the instrument. For example an ammeter with range of 100 A and with 100 divisions on the scale is calibrated normally at four points, say at 25 A, 50 A, 75 A, and 100 A graduation marks, but in practice the instrument may read 60 A; then naturally the user would like to know as to what will be the real value of the current passing through it, when the instrument is reading 60 A. This chapter is mainly based on my research paper [1] published in MAPAN Journal of Metrology Society of India, in 1999.

Another set of instruments are transducer type, which have an arbitrary scale, which is driven by one quantity but depicts a totally different quantity, for example electronic weighing instruments, in which indication will depend upon the electric current through the circuit, but the scale will depict the weight of the body. Proving rings are used to measure force. Force induces linear changes in the diameter of the proving ring, which is measured, but the instrument associated with it indicates force. Same is the case of voltmeter and other electrical measuring instruments etc

In calibration of hydrometers, the author has observed that, sometimes, the correction at the top of the scale was very large and reduced to almost zero at the bottom of the scale. Such results indicate that the length of the scale is too small or too large. In this case there will be a linear relation between correction and indication
of the hydrometer. If the gradient of the linear relation is very small, it suggests that corrections are independent of the scale reading. Similarly the correction assigned to a mercury-in glass thermometer will be a quadratic function of the indication on the thermometer, if diameter of the capillary is uniformly changing.

Normally the number of points at which the instrument is calibrated is very much less than the total number of graduations on it. For finding out the value of input at other points we may like to have some sort of algebraic relation, so that by choosing any numerical value of the independent variable (indication on the scale) the value of the input is obtained. This will enable us to calculate the value of input (dependent variable) for any chosen value of the independent variabl– graduation on the scale

Firstly, we may like to find out if there exists a relation between the corrected input quantities and scale readings. Then we try to establish a graph or an equation of a graph, or an empirical relation between the known inputs and readings taken on the scale of the instrument is required to be given.

Such relations may be a polynomial including a linear relation, a power function, or an exponential function in one variable. A mathematical function becomes specific relation if the values of the constants involved in defining the function are given.

To specify the mathematical relation we use the least square method which gives the best estimates of the constants involved in defining the function. We will find the standard deviation by taking the square root of the average of the squares of the residual errors. This standard deviation is used in calculating the uncertainty of the
estimated input for the given scale reading.

Usually a mathematical relation including the values of constants is given after the calibration of an instrument. But in this chapter, we wish to go one step further.
In addition of the values of constants involved in the function, we find out the uncertainty in assigning the value of the dependent variable (input quantity), by choosing from the function, any value of the independent variable (indication on the scale of the instrument).

First simple case is that in which scale reading indicated by x bears a linear relationship with the standard input quantity

Source Linear calibration function : measurement uncertainty