Q. With suitable examples differentiate between limiting and known errors.
Sol. Limiting Errors (Guarantee Errors): The accuracy and precision of an instrument depends upon its design, the material used and the workmanship that goes into making the instrument. The choice of an instrument for a particular application depends upon accuracy is desired. It is not economical to use expensive materials and skill for the manufacture of the instrument. But and instrument used for an application requiring a high degree of accuracy has to use expensive material and a highly skilled workmanship. The economical production of any instrument requires the proper choice of material, design and skill. In order to assure the purchaser of the quality of the instrument, the manufacture guarantees a certain accuracy. In most instruments the accuracy is guaranteed to be within a certain percentage of the rated value. Thus the manufacture has to specify the deviations from the nominal value of a particular quantity. The limits of these deviations from the specified value are defined as limiting Errors or Guarantee Errors.
We can say that the manufacture guarantees or promises that the error in the item he is selling is no greater than the limit set. The magnitude of a quantity having a nominal value A_{s }and a maximum error or limiting error of ± A must have a magnitude A_{a } between the limits A_{s}-A and A_{s + A }or Actual value of quantity A_{a =} A_{s }± A
For example, the nominal magnitude of a resistor is 100 with a limiting error of ± 10 .
The magnitude of the resistance will be between the limits
Aa =100±10 or Aa≥90 and Aa≤110
In other words the manufacture guarantees that the value of resistance of the resister lies between 90 and 110 .
Example-1 : The value of capacitance of a capacitor is specified as I µF±5% by the manufacturer. Find the limits between which the value of the capacitance is guaranteed.
Solution: The guaranteed value of the capacitance lie within the limits:
Aa = A_{s}(1±)=1*(1±0.05)=0.95to 1.05 µf.
Note: The same idea of a guarantee limiting the worst possible case applies to electrical measurements. The measurements may involve several components, each of which may be delimited by a guarantee error. Thus the same treatment is to be followed for quantities under measurement as is followed for specified quantities.
Example-2 A 0 - 150 V volunteer has a guaranteed accuracy of 1 percent of full scale reading. The voltage measured by this instrument is 75 V. calculate the limiting error in percent.
Solution: The magnitude of limiting error of instrument is .
Combination of Quantities with Limiting Errors: When two or more quantities, each having a limiting error, are combined, it is advantageous to be able to compute the limiting error of the combination. The limiting error can be easily found by considering the relative increment of the function if the final result is in the form of an algebraic equation.
Example-4 : Three resistors have the following ratings:
Determine the magnitude and limiting error in ohm and in percent of the resistance of these resistances connected in series.
Solution : The values of resistances are
The limiting value of resultant resistance
R=(37+75+50)±(1.85+3.75+2.50)=162±8.10O
Magnitude of resistance = 162O and error in ohm =±8.1O.
Percent limiting error of series combination of resistances
Thus we conclude from the above examples from the above examples that the guarantee values are obtained by taking direct sum of the possible errors, adopting the algebraic signs that give the worst possible case. In fact setting of guarantee limits is necessarily a pessimistic process. This is true from manufacturer's view point as regards his promise to the buyer and it is also true of the user in setting accuracy limits in results of lhis measurements.
Probable Error: Let us consider the two points - r and = r. These points are so located that the area bounded by the curve, the x axis and the ordinates erected at x = - r and x = + r is equal to half of the total area under the curve. That is half the deviations lie between x =± r.
A convenient measure of precision is the quantity r. It is called Probable Error or simply P.E. The reason for this name is the fance mentioned above that half the observed values lie between the limits ± r. If we determine r as the result of n measurements and then make an additional measurement, the chances are 50-50 percent that the new value will lie between - r and + r that is, the chances are even that any one reading will have an error no greater than ± r.