Measurement Uncertainty -- Reduction of Systematic Errors

The prerequisite for the reduction of systematic errors is a complete analysis of the measurement system that identifies all sources of error. Simple faults within a system, such as bent meter needles and poor cabling practices, can usually be rectified readily and inexpensively once they have been identified. However, other error sources require more detailed analysis and treatment. Various approaches to error reduction are considered next.

Careful Instrument Design

Careful instrument design is the most useful weapon in the battle against environmental inputs by reducing the sensitivity of an instrument to environmental inputs to as low a level as possible. For instance, in the design of strain gauges, the element should be constructed from a material whose resistance has a very low temperature coefficient (i.e., the variation of the resistance with temperature is very small). However, errors due to the way in which an instrument is designed are not always easy to correct, and a choice often has to be made between the high cost of redesign and the alternative of accepting the reduced measurement accuracy if redesign is not undertaken.

Calibration

Instrument calibration is a very important consideration in measurement systems and therefore calibration procedures are considered in detail in Section 4. All instruments suffer drift in their characteristics, and the rate at which this happens depends on many factors, such as the environmental conditions in which instruments are used and the frequency of their use. Error due to an instrument being out of calibration is never zero, even immediately after the instrument has been calibrated, because there is always some inherent error in the reference instrument that a working instrument is calibrated against during the calibration exercise.

Nevertheless, the error immediately after calibration is of low magnitude. The calibration error then grows steadily with the drift in instrument characteristics until the time of the next calibration. The maximum error that exists just before an instrument is recalibrated can therefore be made smaller by increasing the frequency of recalibration so that the amount of drift between calibrations is reduced.

Fgr. 2

Method of Opposing Inputs

The method of opposing inputs compensates for the effect of an environmental input in a measurement system by introducing an equal and opposite environmental input that cancels it out.

One example of how this technique is applied is in the type of milli-voltmeter shown in Fgr. 2.

This consists of a coil suspended in a fixed magnetic field produced by a permanent magnet.

When an unknown voltage is applied to the coil, the magnetic field due to the current interacts with the fixed field and causes the coil (and a pointer attached to the coil) to turn. If the coil resistance R coil is sensitive to temperature, then any environmental input to the system in the form of a temperature change will alter the value of the coil current for a given applied voltage and so alter the pointer output reading. Compensation for this is made by introducing a compensating resistance R_comp into the circuit, where R_comp has a temperature coefficient equal in magnitude but opposite in sign to that of the coil. Thus, in response to an increase in temperature, R_coil increases but R_comp decreases, and so the total resistance remains approximately the same.

High-Gain Feedback

Fgr. 3

Fgr. 4 --- Feedback device

The benefit of adding high-gain feedback to many measurement systems is illustrated by considering the case of the voltage-measuring instrument whose block diagram is shown in Fgr. 3. In this system, unknown voltage Ei is applied to a motor of torque constant Km, and the induced torque turns a pointer against the restraining action of a spring with spring constant Ks. The effect of environmental inputs on the motor and spring constants is represented by variables Dm and Ds. In the absence of environmental inputs, the displacement of the pointer Xo is given by Xo = KmKsEi. However, in the presence of environmental inputs, both Km and Ks change, and the relationship between Xo and Ei can be affected greatly. Therefore, it becomes difficult or impossible to calculate Ei from the measured value of Xo. Consider now what happens if the system is converted into a high-gain, closed-loop one, as shown in Fgr. 4,byaddingan amplifier of gain constant Ka and a feedback device with gain constant Kf. Assume also that the effect of environmental inputs on the values of Ka and Kf are represented by Da and Df. The feedback device feeds back a voltage Eo proportional to the pointer displacement Xo. This is compared with the unknown voltage Ei by a comparator and the error is amplified. Writing down the equations of the system, we have ...

Because Ka is very large (it is a high-gain amplifier), Kf _ Ka _ Km _ Ks >> 1, and Equation (3) reduces to…

This is a highly important result because we have reduced the relationship between Xo and Ei to one that involves only Kf. The sensitivity of the gain constants Ka, Km, and Ks to the environmental inputs Da, Dm, and Ds has thereby been rendered irrelevant, and we only have to be concerned with one environmental input, Df. Conveniently, it’s usually easy to design a feedback device that is insensitive to environmental inputs: this is much easier than trying to make a motor or spring insensitive. Thus, high-gain feedback techniques are often a very effective way of reducing a measurement system's sensitivity to environmental inputs. However, one potential problem that must be mentioned is that there is a possibility that high-gain feedback will cause instability in the system. Therefore, any application of this method must include careful stability analysis of the system.

Fgr. 5.

Signal Filtering

One frequent problem in measurement systems is corruption of the output reading by periodic noise, often at a frequency of 50 Hz caused by pickup through the close proximity of the measurement system to apparatus or current-carrying cables operating on a mains supply.

Periodic noise corruption at higher frequencies is also often introduced by mechanical oscillation or vibration within some component of a measurement system. The amplitude of all such noise components can be substantially attenuated by the inclusion of filtering of an appropriate form in the system, as discussed at greater length in Section 6. Band-stop filters can be especially useful where corruption is of one particular known frequency, or, more generally, low-pass filters are employed to attenuate all noise in the frequency range of 50 Hz and above.

Measurement systems with a low-level output, such as a bridge circuit measuring a strain-gauge resistance, are particularly prone to noise, and Fgr. 5 shows typical corruption of a bridge output by 50-Hz pickup. The beneficial effect of putting a simple passive RC low-pass filter across the output is shown in Fgr. 5.

Manual Correction of Output Reading

In the case of errors that are due either to system disturbance during the act of measurement or to environmental changes, a good measurement technician can substantially reduce errors at the output of a measurement system by calculating the effect of such systematic errors and making appropriate correction to the instrument readings. This is not necessarily an easy task and requires all disturbances in the measurement system to be quantified. This procedure is carried out automatically by intelligent instruments.

Intelligent Instruments

Intelligent instruments contain extra sensors that measure the value of environmental inputs and automatically compensate the value of the output reading. They have the ability to deal very effectively with systematic errors in measurement systems, and errors can be attenuated to very low levels in many cases. A more detailed coverage of intelligent instruments can be found in Section 11.

NEXT: Quantification of Systematic Errors

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