Static Characteristics

Static Characteristics

AU : May−06,11,13,14,15,16,18,19, Dec.−04,08,09,ll,12,14,15

As mentioned earlier, the static characteristics are defined for the instruments which measure the quantities which do not vary with time. The various static characteristics are accuracy, precision, resolution, error, sensitivity, threshold, reproducibility, zero drift, stability and linearity.

1. Accuracy

It is the degree of closeness with which the instrument reading approaches the true value of the quantity to be measured. It denotes the extent to which we approach the actual value of the quantity. It indicates the ability of instrument to indicate the true value of the quantity. The accuracy can be expressed in the following ways : −

1) Accuracy as 'Percentage of Full Scale Reading' : − In case of instruments having uniform scale, the accuracy can be expressed as percentage of full scale reading. For example, the accuracy of an instrument having full scale reading of 50 units may be expressed as ± 0.1 % of full scale reading. From this accuracy indication, practically accuracy is expressed in terms of limits of error. So for the accuracy limits specified above, there will be ± 0.05 units error in any measurement. So for a reading of 50 units, there will be error of ± 0.05 units i.e.± 0.1 % while for a reading of 25 units, there will be error of ± 0.05 units in the reading i.e. ± 0.2 %. Thus as reading decreases, error in measurement is ± 0.05 units but net percentage error is more. Hence, specification of accuracy in this manner is highly misleading.

2) Accuracy as 'Percentage of True Value':− This is the best method of specifying the accuracy. It is to be specified in terms of the true value of quantity being measured. For example, it can be specified as ± 0.1 % of true value. This indicates that in such cases, as readings get smaller, error also gets reduced. Hence accuracy of the instrument is better than the instrument for which it is specified as percent of full scale reading.

3) Accuracy as 'Percentage of Scale Span':− For an instrument, if amax is the maximum point for which scale is calibrated, i.e. full scale reading and amin is the lowest reading on scale. Then (a_max − a_min) is called scale span or span of the instrument. Accuracy of the instrument can be specified as percent of such scale span. Thus for an instrument having range from 25 units to 225 units, it can be specified as ± 0.2 % of the span i.e. ± [(0.2/100) × (225 − 25)] which is ± 0.4 units error in any measurement.

4) Point Accuracy :− Such an accuracy is specified at only one particular point of scale. It does not give any information about the accuracy at any other point on the scale. The general accuracy of an instrument cannot be specified, in this manner. But the general accuracy can be specified by providing a table of the point accuracy values calculated at various points throughout the entire range of the instrument.

Thus the accuracy, in whatever way it may be specified, gives the equipment accuracy with a particular set up and other conditions and does not include any personal accuracy.

 

2. Precision

It is the measure of consistency or repeatability of measurements.

Key Point It denotes the closeness with which individual measurements are departed or distributed about the average of number of measured values.

Let us see the basic difference between accuracy and precision. Consider an instrument on which, readings upto 1/1000 th of unit can be measured. But the instrument has large zero adjustment error. Now every time reading is taken, it can be taken down upto 1/1000 of unit. So as the readings agree with each other, we say that the instrument is highly precise. But, though the readings are precise upto 1/1000 th of unit, the readings are inaccurate due to large zero adjustment error. Every reading will be inaccurate, due to such error. Thus a precise instrument may not be accurate.

Thus the precision means sharply or clearly defined and the readings agree among themselves. But there is no guarantee that readings are accurate.

An instrument having zero error, if calibrated properly, can give accurate readings  but in that case still, the readings can be obtained down upto 1/1000 th of unit only. Thus accuracy can be improved by calibration but not the precision of the instrument.

Consider another example. A known weight of 100 grams is measured by an instrument. Five times, the weight has been recorded. The readings obtained are 103, 104, 105, 103, 105. The average indicated value is 104 grams. Hence the maximum deviation from the average reading is ±1 gram in 100 grams actual value. Thus, the scale of the instrument can be calibrated to read ±1 gram. But what about the accuracy ? The readings are not accurate. The accuracy of the instrument is only (105 − 100/100)% i.e. 5 % . Thus there is a precision of ± 1 % but the accuracy is only 5 %.

Key Point This confirms the fact that high degree of precision does not guarantee the accuracy. It is the accurate calibration that makes the accurate measurement possible.

The precision is composed of two characteristics:−

• Conformity and

• Number of significant figures.

 

a. Conformity

Consider a resistor having true value as 2385692 Ω, which is being measured by an ohmmeter. Now, the meter is consistently measuring the true value of the resistor. But the reader, can read consistently, a value as 2.4 M Ω due to nonavailability of proper scale. The value 2.4 M Ω is estimated by the reader from the available scale. There are no deviations from the observed value. The error created due to the limitation of the scale reading is a precision error.

The example illustrates that the conformity is necessary, but not sufficient condition for precision. Similarly, precision is necessary but not the sufficient condition for accuracy.

Key Point An accurate instrument should be precise but a precise instrument may not be accurate.

 

Significant Figures

The precision of the measurement is obtained from the number of significant figures, in which the reading is expressed. The significant figures convey the actual information about the magnitude and the measurement precision of the quantity.

For example, a resistance of 110 Ω, specified by an instrument may be closer to 109 Ω or 111 Ω. Thus there are 3 significant figures. While if it is specified as 110.0 Ω then it may be closer to 110.1 Ω or 109.9 Ω. Thus there are now 4 significant figures.

Key Point Thus more the significant figures, the greater is the precision of measurement.

Number of times, the large numbers with zeros before a decimal point are used to approximate populations or the amounts of money. For example, the price of a vehicle is reported as ₹ 450,000. This means the true value of the vehicle lies between ₹ 449,999 and ₹ 450,001. Thus, there are six significant figures. But what is the meaning of the reported price is, it is closer to ₹ 450,000 rather than ₹ 440,000 or ₹ 460,000. In this case, there are only two significant figures. To avoid this confusion, the large numbers are expressed in a scientific notation using the powers of ten. For example, the price of ₹ 450,000 must be expressed as 4.5 × 10⁵. Thus now, there are only two significant figures. The uncertainty due to the zeros to the left of the decimal point is usually resolved by such scientific notation.

The precision can be mathematically expressed as

P = Precision

X_n = Value of n^th measurement

 = Average of the set of measured values

 

 

 

 

Example 1.5.1 The table shows the set of 5 measurements recorded in a laboratory. Calculate the precision of the 3^rd measurement.

PPPPPPPPPPPP

Solution :

The average value for the set of measurements is,

PPPPPPPPPPPP

The value of 3^rd measurement is X_n = 52 where n = 3

 PPPPPPPPPPP

This is the precision of the 3^rd measurement.

 

 

3. Error

The most important static characteristics of an instrument is its accuracy, which is generally expressed in terms of the error called static error.

Key Point The algebraic difference between the indicated value and the true value of the quantity to be measured is called an error.

Mathematically it can be expressed as, .

e = A_t − A_m

where e = error

A_m = measured value of the quantity

A_t = true value of the quantity

In this expression, the error denoted as e is also called absolute error. The absolute error does not indicate precisely the accuracy of the measurements. For example, absolute error of ± 1 V is negligible when the voltage to be measured is of the order of 1000 V but the same error of ± 1 V becomes significant when the voltage under measurement is 5 V or so. Hence, generally instead of specifying absolute error, the relative or percentage error is specified.

Mathematically, the relative error can be expressed as,

PPPPPPPPPP

The percentage relative error is expressed as,

PPPPPPPPPP

From the relative percentage error, the accuracy can be mathematically expressed as, 

PPPPPPPP

Where A = relative accuracy

and a = A × 100 %

Where a = percentage accuracy

The error can also be expressed as a percentage of full scale reading as,

Error as a percentage of full scale reading = PPPPPPP

where f.s.d. = Full scale deflection.

Example 1.5.2 The expected value of the voltage to be measured is 150 V. However, the measurement gives a value of 149 V. Calculate (i) Absolute error; (ii) Percentage error; (Hi) Relative accuracy; (iv) Percentage accuracy and (v) Error expressed as percentage of full scale reading, if the scale range is 0 − 200 V.

Solution :

The expected value means true value, A_t = 150 V

The measured value is given as 149 V.

∴ A_m = 149 V

i) e = absolute error =  A_t − A_m = 150 − 149 = + 1 V

ii) PPPPPPP

iii) PPPPPPP

iv) % a = A x 100 = 0.9933 x 100 = 99.33 %

v) % error expressed as percentage of full scale reading is,

PPPPPPPPPPP  as f.s.d. is 200 V = 0.5 %

 

4. Sensitivity

The sensitivity denotes the smallest change in the measured variable to which the instrument responds.

It is defined as the ratio of the changes in the output of an instrument to a change in the value of the quantity to be measured. 

Mathematically it is expressed as,

∴ Sensitiviy = infinitesimal change in output/infinitesimal change in input  PPPPPP

∴ Sensitiviy = ∆q₀/∆q_i      PPPPP

Thus, if the calibration curve is linear, as shown in the Fig. 1.5.1 (a), the sensitivity of the instrument is the slope of the calibration curve.

If the calibration curve is not linear as shown in the Fig. 1.5.1 (b), then the sensitivity varies with the input.

 PPPPPPPPPPP

The sensitivity is always expressed by the manufacturers as the ratio of the magnitude of quantity being measured to the magnitude of the response. Actually, this definition is the reciprocal of the sensitivity is called inverse sensitivity or deflection factor. But manufacturers call this inverse sensitivity as a sensitivity.

Inverse sensitivity = deflection factor

∴ Deflection factor = 1/ sensitivity = ∆q_i/∆q₀      PPPPPPPPPPP

The units of the sensitivity are millimeter per micro−ampere, millimeter per ohm, counts per volt, etc. while the units of a deflection factor are micro-ampere per millimeter, ohm per millimeter, volts per count, etc.

The sensitivity of the instrument should be as high as possible and to achieve this the range of an instrument should not greatly exceed the value to be measured.

Example 1.5.3 A particular ammeter requires a change of 2 A in its coil to produce a change in deflection of the pointer by 5 mm. Determine its sensitivity and deflection factor. 

Solution : The input is current while output is deflection.

Sensitivity = change in output/         change in input = 5mm/2A = 2.5 mm /A  PPPPPPP

Deflection factor = 1/ sensitivity = 1/2.5 =  0.4 A / mm   PPPPPP

 

5. Resolution

 

 

It is the smallest increment of quantity being measured which can be detected with certainity by an instrument.

Key Point  Thus, the resolution means the smallest measurable input change.

So if a non zero input quantity is slowly increased, output reading will not increase until some minimum change in the input takes place. This minimum change which causes the change in the output is called resolution. The resolution of an instrument is also referred to as discrimination of the instrument. The resolution can affect the accuracy of the measurement.          .

Example 1.5.4 A 30 cm scale has 30 uniform divisions. l/20 th of a scale division can be estimated with a fair degree of certainty. Determine the resolution of the scale in mm.

 Solution : 1 scale division = full scale deflection/number of division = 30cm/30 = 1cm = 10 mm

Resolution = 1/20 × scale division = 1/20 × (10 mm) = 0.5 mm

 

 

 6. Theshold

If the input quantity is slowly varied from zero onwards, the output does not change until some minimum value of the input is exceeded. This minimum value of the input is called threshold.

Key Point  Thus, the resolution is the smallest measurable input change while the threshold is the smallest measurable input.

 

7. Linearity

The instrument requires the property of linearity that is the output varies linearly, according to the input. The linearity is defined as the ability to reproduce the input characteristics symmetrically and linearly. Graphically such relationship between input and output is represented by a straight line.

The graph of output against the input is called the calibration curve.

Key Point  The linearity property indicates the straight line nature of the calibration curve.

The linearity is defined as the maximum deviation of the actual calibration curve (output) from the idealized straight line, expressed as a percentage of full scale reading or a percentage of the actual reading.

The Fig. 1.5.2 shows the actual calibration curve and idealized straight line.

 PPPPPPPPPP

Thus, the linearity is defined as,

PPPPPPPPPPPP

It is desirable to have an instrument as linear as possible as the accuracy and linearity are closely related to each other.

 

8. Zero Drift

The drift is the gradual shift of the instrument indication, over an extended period during which the value of the input variable does not change.

The zero drift is defined as the deviation in the instrument output with time, from its zero value, when the variable to be measured is constant. The whole instrument calibration may gradually shift by the same amount.

There are many environmental factors which affect the drift. These factors are stray electric field, stray magnetic field, temperature changes, contamination of metal, changes in the atomic structure, mechanical vibrations, wear and tear, corrosion, etc.

The drift is undesirable and cannot be easily compensated for. It must be carefully guarded against by continuous inspection. 

 

9. Reproducibility

It is the degree of closeness with which a given value may be repeatedly measured. It may be specified in terms of units for a given period of time.

The perfect reproducibility indicates no drift in the instrument.

The repeatability is denied as variation of scale reading ans is random in nature. Both reproducibility and the repeatability are a measure of the closeness with which a given input may be measured again and again. The Fig shows the input and output relationship with positive and negative repeatability.

PPPPPPPPPPPPPP

 

10. Stability

The ability of an instrument to retain its performance throughout its specified operating life and the storage life is defined as its stability.

 

 

 11. Tolerance

The maximum allowable error in the measurement is specified interms of some value which is called tolerance. This is closely related to the accuracy.

Actually tolerance is not the static characteristics of measuring instrument but it is mentioned because in some instruments the accuracy is specified interms of tolerance values.

Key Point  The tolerance indicates the maximum allowable deviation of a manufactured component from a specified value.

 

12. Range or Span

The minimum and maximum values of a quantity for which an instrument is designed to measure is called its range or span. Sometimes the accuracy is specified in terms of range or span of an instrument.

 

13. Bias

The constant error which exists over the full range of measurement of an instrument is called bias. Such a bias can be completely eliminated by calibration. The zero error is an example of bias which can be removed by calibration. 

 

14. Hysteresis

If the input to the instrument is increased from a negative value, the output also increases. This is shown by curve 1 in the Fig. 1.5.4. But if the input is now decreased steadily, the output does not follow the same curve but lags by certain value. It traces the curve 2 as shown in the Fig. 1.5.4. The difference between the two curves is called hysteresis. The maximum input hysteresis and the maximum output hysteresis are shown in the Fig. 1.5.4. These are generally expressed as the percentage of the full scale reading.

PPPPPPPPPPPP

 

15. Dead Space

In some instruments, it is possible that till input increases beyond certain value, the output does not change. So for certain range of input values there is no change in output. This range of input is called dead space. This is shown in the Fig. 1.5.4. There is possibility that instrument without hysteresis may show the dead space in their output characteristics. Backlash in gears is a good example which causes the dead space.

 

16. Span Drift or Sensitivity Drift

If there exists a proportional change in the indication, all along the upward scale then the drift from nominal characteristics is called span drift or sensitivity drift.

The characteristics with span drift are shown in the Fig. 1.5.5.

PPPPPPPPPPP

Fig. 1.5.5 Span drift

 

Example 1.5.5 A voltmeter reads 111.5 V. The error taken from an error curve is 5.3 %. Find the true value of the voltage.

Solution :   A_m = 111.5 V, % e = 5.3 %

PPPPPPPPPPP

∴ 0.053 A_t = A_t − 111.5 i.e. A_t = 117.74 V

Example 1.5.6

A 0 − 100 V voltmeter has 200 scale divisions which can be read to 1/2 division. Determine the resolution of the meter in volt.

Solution :

1 scale division = Full scale division/ Number of divisions = 100/200 = 0.5 V  PPPPP

∴ Resolution = 1/2 × scale division  = 1/2 × 0.5 = 0.25 V   PPPPPP

Example 1.5.7

For the network shown in the Fig. 1.5.6, find the voltage reading on voltmeter, if voltmeter sensitivity is 1 kΩ/volt. If the voltmeter is replaced by another voltmeter having sensitivity 25 kΩ/volt, find the new reading. Comment on the answer.

PPPPPPPPPPP

Solution:

PPPPPPPPPPPP

By voltage divider, the voltage across the points A − B is,

V_AB = 50 kΩ × 150 /(100 K + 50 K) =  50 V  PPPPPP

Now voltmeter one has sensitivity 1 kΩ/volt, hence resistance offered by the voltmeter is,

R = (1 kΩ/volt) × 50 = 50 kΩ

Hence circuit becomes,

∴ V_AB = (50 kΩ || 50 kΩ) x-PPPPPPPPPPPP

Thus the percentage error in the reading is,

% e = true value − measured value/ true value × lOO =   (50 − 30) /50 × lOO = + 40 %  PPPPPPPPPP

Now the voltmeter is replaced by another one having sensitivity 25 kΩ/volt. Thus it will offer the resistance,

PPPPPPPPPPPPP

R = 25 kΩ/volt  × 50 = 1250 kΩ

Hence circuit becomes,

∴ V_AB = (50 kΩ || 1250 kΩ)  PPPPPPPPPPP

= 48.077 × 150/(100 + 48.077) = 48.701 V PPPPPP

Thus the percentage error in the reading is,

% e = (50 – 48.701)/50 × 100 = + 2.59 % PPPPPP

Key Point

Thus the voltmeter with low sensitivity shows more error while the voltmeter with high sensitivity shows less error.

Example 1.5.8

A voltmeter having a sensitivity of 1.5 kΩ/volt reads 80 V on its 150 V range, when connected across an unknown resistor in series with a milliammeter. The ammeter reads 15 mA. Calculate

i) Apparent resistance

ii) Actual resistance of unknown resistor

iii) Error due to loading effect of voltmeter

iv) Percentage relative accuracy.

Solution :

PPPPPPPPP

The circuit diagram is shown in the Fig. 1.5.7.

The total circuit resistance, neglecting resistance of milliammeter is,

R_T = V/I =   80/15 × 10⁻³ = 5.333 kΩ  PPPPPPPPP

i)Thus the appearent value of the resistance is

R_app = 5.333 kΩ

ii) Let us calculate the actual R_x.

The resistance of the voltmeter be R_v.

∴ R_v = 1.5 kΩ/volt × 150 as 150 V is full scale reading = 225 kΩ

Thus PPPPPPPPPPPPP

∴ R_x + 225 = 42.19 R_x   i.e.   R_x = 4.462 kΩ

This is the actual value of the unknown resistance.

iii) PPPPPPPP

iv) The relative accuracy.

% A = (1 − |error|)  × 100 = (1 – 0.0236) × 100 = 97.63 %

Example 1.5.9

A step input of 5 A is applied to an ammeter. The pointer swings to 5.18 A and finally comes to rest at 5.02 A.

a) Determine the overshoot of the reading in amperes in percentage of final reading.

b) Determine the percentage of error in the instrument.   

AU: Dec. - 04

Solution:

The overshoot of 5.18 − 5.02 = 0.16 A when final reading is 5.02 A.

a) % overshoot = 0.16/5.02 × 100 =  3.187 %  ppppppp

b) % e = (A_t − A_m)/A_t  × 100 = (5 − 5.02)/5 × 100 = −0.4 %  PPPPP

Review Questions

1. Define any three parameters of the static characteristics of instruments.

AU: May -  06, Dec -18, Marks 6

2. Explain the static characteristics of a measurement system, with examples.

AU: May-11, 13, 15, 16, 18, 19, Dec.-08, 09, 11, 12, 14, 15, Marks 16

3. A moving coil voltmeter has a uniform scale with 100 divisions, the full scale reading is 200 V and 1/10 of scale division can be estimated with a fair degree of certainity. Determine the resolution of the instrument in volt.

[Ans. : 0.2 V]

4. A digital voltmeter has a read out range from 0-9999 counts. Determine the resolution of the instrument in volt when the full scale reading is 9.999 V.

[Ans. : 1 mV]

5. A true value of voltage across resister is 50 V. The instrument reads 49 V. Calculate

i) Absolute error

ii) Percentage error

iii) Percentage accuracy

[Ans. : 1 V, 2%, 98%]

6. Define accuracy and reproducibility of an instrument and explain.

AU: May-14, Marks 6