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GENERALIZED PERFORMANCECHARACTERISTICS OFINSTRUMENTSLecture 6Instructor : Dr Alivelu M Parimi


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Static Characteristics Accuracy, Precision, Range and Span, Resolution and Threshold, Sensitivity , Linearity, Drift and Hysteresis STATISTICAL ANALYSIS OF RANDOM ERRORS

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Linearity

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Static Characteristics: Driftand Hysteresis.

Drift is a complex phenomenon for which the observed effects arethat the sensitivity and offset values vary.

Drift may be classified into three categories: Zero drift : If the whole calibration gradually shifts due to the

slippage, permanent set, or due to undue warming up ofelectronic tube circuits, zero drift sets in. This can beprevented by zero setting.

Span drift or sensitivity drift : If there is proportional change inthe indication all along the upward scale, the drift is calledspan drift or sensitivity drift.

Zonal drift : In this case the drift occurs only over a span of aninstrument, then it is called zonal drift.

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Classification of Drift


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Example

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A spring balance is calibrated in anenvironment of 20 degree centigrade and hasthe following deflection-load (mm vs kg)characteristics

Load (kg) 0 1 2 3

Deflection in cm at 20 0C 0 20 40 60

Deflection in cm at 30 0C 5 27 49 71

Find the effect of temperature on Sensitivityand Zero drift


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Solution

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Solution

Zero drift =C

cm

305

= 0.166 cm/C

Sensitivity at 20 oC =kg

mm203

60

Sensitivity at 30 oC = 3

5-71

kgmm

= 22kg

mm

Sensitivity drift = C10

kg/mm20-22 = 0.2C

kg/mm


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Static Characteristics:hysteresis

Hysteresis is the magnitude of error caused in the output for agiven value of input when the value is approached fromopposite directions (increasing/decreasing).

The causes of hysteresis are backlash, elastic deformation,magnetic characteristics, friction effects etc.

This is commonly caused in mechanical instruments by loose

gears and linkages and friction.

Hysteresis effect is prominent in magnetization anddemagnetization phenomena. 9


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STATISTICAL ANALYSIS OF

RANDOM ERRORS Any given instrument is prone to errors either due to aging or

due to manufacturing tolerances.

Every experimental result is subject to errors which will creepinto regardless of the care taken.

One can attempt to minimize errors but cannot eliminatethem completely.

Some form of analysis must be performed on all experimentaldata.

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Terms used in Statistical Analysis

Arithmetic mean

If x1, x2, x3. are the n readings or samples of the measured variable,the arithmetic mean is given by:

Median Geometric mean= n-th root of (x 1)(x 2)...(x n) Deviation

Deviation is departure of the observed reading from the arithmeticmean of the group. Let deviation of reading x 1 be d 1 and that ofreading x 2 be d 2 etc.

d i = xi x Average deviation

Standard deviation 12

n x x x x

x x nm....321

x

n

d in

1i

n

d

n

x x N

i

i

N

i 1

2

1

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Example Consider the data shown in Table obtained by measuring the

mass of a metal strip 24 times with the same instrument Find the mean and standard deviation

Mean value=8.148 gstandard deviation=0.004

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Readings of 24 Measurements of Mass of Metal Strip


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Normal or Gaussian curve oferrors

The reproducibility of this set of measurements can bevisualized by plotting the data as a histogram which showshow often (frequency) a given value was obtained in the set.

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Normal or Gaussian curve oferrors

If we were to conduct a very large number of measurementson the metal strip, we would have obtained a histogramwhose shape resembles a bell

15Normal Distribution Curve Obtained by Large Number ofMeasurements

This bell-shaped curve denotes the

normal probability distribution for alarge number of massmeasurements


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Normal or Gaussian curve of errors The exact shape of the normal distribution is characterized by

two parameters: the mean value , and the standard deviation .

For a large number of measurements, the mean value is alsothe most probable value, as shown on the plot .

The standard deviation is a measure of the breadth of thecurve; the larger the standard deviation, the broader thedistribution.

Put differently, the less precise the measurement, the broader

the distribution and the larger the standard deviation.

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Gaussian distribution

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Table 3.2: Values of Area under Gaussuain Distribution Curve

Table list areas under the

curve between zero andvarious values of z

This table will have areasfrom 0 to maximum 0.5


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Example

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Entries in the Table 3.2 correspond to area under the normal curve. With z = 0 area is zero

and as z increases area increases, maximum area is 0.5. Suppose we have to know within

hat deviation from mean, 50% of the readings will lie, then in table we will look for only

0.25 ( since the curve is symmetric). The value of area of 0.25 lies for value of z between

0.67 and 0.68 (exact is 0.6745).

6745.0xxor xx

6745.0z

If we had table giving area from - and particular value of z, then find value of z, say z 1 for

area of 0.75, then find z 2 for area 0.25, z 1 z2 will be 0.6745.

If it has to be found that what percent of readings would fall within + of mean, i.e.

deviation 1zsoxx

For z = 1 f(z) = 0.3413


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Example A value of R = 92.2 + 0.1 ohms (where 0.1 ohm is the standard

deviation) is specified for a batch of 1000 resistors. How manyare estimated to have values in the rangeR = 92.2 + 0.15 ohms? Assume normal distribution.

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Solution

Given deviation x = 0.15 ohms;

x x z

=

1.015.0

= 1.5

Corresponding to 1.5, the area under the Gaussian curve is 0.4332

Therefore the probable number of resistors having a value of 92.2 + 0.15 is1000)0.4332(2 = 866


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