Fundamentals of Measurement

Measurement is the process of comparing an unknown quantity with a standard of the same kind. The methods are grouped by how the comparison is made.

Fundamental methods

• Direct method: the unknown is compared directly with the standard or read on a calibrated scale. Example: length with a steel rule, mass on a beam balance.
• Indirect method: the quantity is found by measuring other quantities related to it by a known law. Example: power from $P = VI$, density from mass and volume.
• Comparison method: the unknown is compared with a known quantity of the same kind (a balance with standard weights).
• Substitution method: the unknown is replaced by a known standard that gives the same effect.
• Null method: the known quantity is adjusted until the difference is zero.

Deflection and null methods

Example of null method: in a Wheatstone bridge the variable arm is adjusted until the galvanometer shows zero, and the unknown resistance is $R_x = R_2 R_3/R_1$.

Example of deflection method: a spring balance shows weight from the spring extension on a marked scale.

A generalized measurement system converts the measured quantity (measurand) into a form suitable for observation. It has three main stages: sensing/transducing, signal conditioning, and output (readout).

 Measurand    +-----------+   +-------------+   +----------+
 ----------->| Sensor /  |-->|   Signal    |-->| Terminal |
 (temperature)| transducer|   | conditioning|   | (readout)|
              +-----------+   +-------------+   +----------+
   Stage I        Stage II (modify, amplify)      Stage III
                         ^
                    Power supply / calibration

Functions of the stages

1. Sensor-transducer stage (detector): senses the measurand and gives an output related to it. It should respond only to the measurand and not load the source.
2. Signal-conditioning stage: modifies the transducer output into a suitable form. It includes amplification, filtering, bridge circuits, analog-to-digital conversion, linearization and modulation.
3. Terminating (readout) stage: presents the result to the observer or controller. It may be an indicator, recorder, display or data logger.

Thermocouple temperature indicator

• Sensor: the thermocouple produces a small emf (mV) proportional to the difference between hot and cold junction temperatures.
• Signal conditioning: a cold-junction compensation circuit adds the reference emf, an amplifier raises the mV signal, a filter removes noise, and linearization corrects the nonlinear emf-temperature relation. An ADC converts it to digital form.
• Readout: a digital display or chart recorder shows the temperature in $^\circ$C.

Calibration against a known standard temperature links the readout to the measurand.

Power and its limiting error

Given: $V = 120$ V ($\pm 0.5\%$), $R = 50\ \Omega$ ($\pm 1\%$).

$$P = \frac{V^2}{R} = \frac{120^2}{50} = 288\ \text{W}$$

For a quantity $P = V^2 R^{-1}$, limiting (worst-case) relative errors add, each multiplied by its power:

$$\frac{\delta P}{P} = 2\,\frac{\delta V}{V} + \frac{\delta R}{R} = 2(0.5) + 1 = 2\%$$ $$\delta P = 0.02 \times 288 = 5.76\ \text{W}$$

Answer: $P = 288 \pm 5.76$ W (limiting error $\pm 2\%$).

If the errors are independent, the probable error is $\sqrt{(2\times 0.5)^2 + 1^2} = 1.41\%$, about $\pm 4.1$ W.

Systematic and random errors