Static Characteristics of Measurement System

(a) Accuracy and precision

Accuracy is the closeness of a measured value to the true value. Precision is the closeness of repeated readings of the same quantity to each other (repeatability).

Example: a target shot. Shots all clustered far from the bullseye are precise but not accurate. Shots scattered evenly around the bullseye are accurate on average but not precise. A voltmeter reading 4.98, 4.99, 4.98 V for a true 5.50 V is precise but inaccurate.

(b) Definitions

• Sensitivity: ratio of the change in output to the change in input, $S = \Delta q_o/\Delta q_i$ (slope of the calibration curve). Example: 5 mV/bar.
• Threshold: the smallest input change from zero that produces a detectable output.
• Resolution: the smallest change in input (from a non-zero value) that the instrument can detect. For a digital meter it is one least-count digit.
• Tolerance: the maximum permitted departure from a specified value, usually given as $\pm$ value or $\pm\%$; it is the permissible error, not the instrument's inherent accuracy.

(a) Hysteresis

Hysteresis is the difference in output for the same input depending on whether the input is increasing or decreasing. It is caused by friction, backlash, elastic after-effect and magnetic hysteresis. It is quoted as the maximum difference between the up-scale and down-scale outputs, as a percentage of full-scale output.

 Output |           ___--- up
        |       _--/ _--- down
        |    _-/ _--/   <- hysteresis band
        |  _/ _-/
        | /_-/
        +------------------ Input

(b) Dead space

Dead space (dead band) is the range of input over which there is no change in output, even though the input changes, usually because of backlash, friction or a preload. The instrument responds only after the input leaves this band. Hysteresis is not the same: it exists at all points, while dead space is a flat zone.

(c) Linearity

Linearity is the closeness of the calibration curve to a straight line. Nonlinearity is the maximum deviation from the reference line, expressed as % of full-scale output. The reference line can be defined in three ways.

• Terminal (end-point) linearity: a line joining the zero and full-scale points. It is simple but gives large deviations.
• Independent linearity: two parallel lines enclose the curve with minimum spacing; the line midway is the reference. Nonlinearity is half the spacing.
• Least-squares linearity: the line that minimizes $\sum (y_i - mx_i - c)^2$. It gives the smallest overall error and is the most used.

 Output |        . . .  actual curve
        |     .  /  <- best straight line
        |   . /
        | ./
        +----------------- Input

Method: fit $y = mx + c$ by least squares, then find the largest difference between data and the line.

Least-squares fit

For $n = 6$ points: $\sum x = 30$, $\sum y = 15.04$, $\sum x^2 = 220$, $\sum xy = 149.57$ (units bar, V).

$$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{6(149.57) - 30(15.04)}{6(220) - 900} = \frac{446.22}{420} = 0.4986\ \text{V/bar}$$ $$c = \frac{\sum y - m\sum x}{n} = \frac{15.04 - 0.4986(30)}{6} = 0.0138\ \text{V}$$

So the line is $y = 0.4986\,x + 0.0138$.

Nonlinearity

Maximum deviation $= 0.0419$ V at 4 bar.

Full-scale output $= 5.01 - 0.02 = 4.99$ V.

$$\text{Nonlinearity} = \frac{0.0419}{4.99}\times 100 = 0.84\%\ \text{of FSO}$$

Answer: sensitivity $= 0.499$ V/bar, zero offset $= 0.0138$ V, maximum nonlinearity $= 0.84\%$ of FSO.

(a) Sensitivity to disturbance

Environmental inputs such as ambient temperature, supply voltage, vibration or humidity can change the output of an instrument even when the measurand is constant. These are called disturbances (interfering and modifying inputs). Two effects are described.

• Zero drift (zero shift): the whole calibration line moves up or down by a constant amount, so the output at zero input changes. It is quoted per unit disturbance, e.g. mV/$^\circ$C.
• Sensitivity drift (scale-factor drift): the slope of the calibration line changes, so the error grows with input. Quoted as, e.g., (mV/bar) per $^\circ$C.

 Output |      /  <- sensitivity drift (slope change)
        |    //
        |  ///_/  <- zero drift (parallel shift)
        |_/_/
        +---------------- Input

(b) Numerical

Temperature change $\Delta T = 40 - 20 = 20\ ^\circ$C.

Nominal output $= 5 \times 6 = 30$ mV.

Zero drift $= 0.2 \times 20 = 4$ mV.

Sensitivity drift effect $= (0.01\times 20)\times 6 = 1.2$ mV.

$$V_o = 30 + 4 + 1.2 = 35.2\ \text{mV}$$

Answer: output $= 35.2$ mV, an error of $+5.2$ mV (17%) due to temperature.