Measurement Uncertainty -- Sources of Systematic Error

The main sources of systematic error in the output of measuring instruments can be summarized as:

• effect of environmental disturbances, often called modifying inputs

• disturbance of the measured system by the act of measurement

• changes in characteristics due to wear in instrument components over a period of time

• resistance of connecting leads

These various sources of systematic error, and ways in which the magnitude of the errors can be reduced, are discussed here.

System Disturbance due to Measurement

Disturbance of the measured system by the act of measurement is a common source of systematic error. If we were to start with a beaker of hot water and wished to measure its temperature with a mercury-in-glass thermometer, then we would take the thermometer, which would initially be at room temperature, and plunge it into the water. In so doing, we would be introducing a relatively cold mass (the thermometer) into the hot water and a heat transfer would take place between the water and the thermometer. This heat transfer would lower the temperature of the water. While the reduction in temperature in this case would be so small as to be undetectable by the limited measurement resolution of such a thermometer, the effect is finite and clearly establishes the principle that, in nearly all measurement situations, the process of measurement disturbs the system and alters the values of the physical quantities being measured.

A particularly important example of this occurs with the orifice plate. This is placed into a fluid-carrying pipe to measure the flow rate, which is a function of the pressure that is measured either side of the orifice plate. This measurement procedure causes a permanent pressure loss in the flowing fluid. The disturbance of the measured system can often be very significant.

Thus, as a general rule, the process of measurement always disturbs the system being measured.

The magnitude of the disturbance varies from one measurement system to the next and is affected particularly by the type of instrument used for measurement. Ways of minimizing disturbance of measured systems are important considerations in instrument design. However, an accurate understanding of the mechanisms of system disturbance is a prerequisite for this.

Measurements in electric circuits

Fgr. 1

In analyzing system disturbance during measurements in electric circuits, Thevenin's theorem is often of great assistance. For instance, consider the circuit shown in Fgr. 1a in which the voltage across resistor R5 is to be measured by a voltmeter with resistance Rm. Here, Rm acts as a shunt resistance across R5, decreasing the resistance between points AB and so disturbing the circuit. Therefore, the voltage Em measured by the meter is not the value of the voltage Eo that existed prior to measurement. The extent of the disturbance can be assessed by calculating the open-circuit voltage Eo and comparing it with Em.

The ´venin's theorem allows the circuit of Fgr. 1a comprising two voltage sources and five resistors to be replaced by an equivalent circuit containing a single resistance and one voltage source, as shown in Fgr. 1b. For the purpose of defining the equivalent single resistance of a circuit by Thevenin's theorem, all voltage sources are represented just by their internal resistance, which can be approximated to zero, as shown in Fgr. 1c. Analysis proceeds by calculating the equivalent resistances of sections of the circuit and building these up until the required equivalent resistance of the whole of the circuit is obtained. Starting at C and D, the circuit to the left of C and D consists of a series pair of resistances (R1 and R2)in parallel with R3, and the equivalent resistance can be written as ...

Moving now to A and B, the circuit to the left consists of a pair of series resistances (RCD and R4) in parallel with R5. The equivalent circuit resistance RAB can thus be written as ....

Substituting for RCD using the expression derived previously, we obtain ....

Defining I as the current flowing in the circuit when the measuring instrument is connected to it, we can write....and the voltage measured by the meter is then given by ...

In the absence of the measuring instrument and its resistance Rm, the voltage across AB would be the equivalent circuit voltage source whose value is Eo. The effect of measurement is therefore to reduce the voltage across AB by the ratio given by ....

It’s thus obvious that as Rm gets larger, the ratio Em/Eo gets closer to unity, showing that the design strategy should be to make Rm as high as possible to minimize disturbance of the .... measured system. (Note that we did not calculate the value of Eo, as this is not required in quantifying the effect of Rm.) At this point, it’s interesting to note the constraints that exist when practical attempts are made to achieve a high internal resistance in the design of a moving-coil voltmeter. Such an instrument consists of a coil carrying a pointer mounted in a fixed magnetic field. As current flows through the coil, the interaction between the field generated and the fixed field causes the pointer it carries to turn in proportion to the applied current (for further details, see Section 7). The simplest way of increasing the input impedance (the resistance) of the meter is either to increase the number of turns in the coil or to construct the same number of coil turns with a higher resistance material. However, either of these solutions decreases the current flowing in the coil, giving less magnetic torque and thus decreasing the measurement sensitivity of the instrument (i.e., for a given applied voltage, we get less deflection of the pointer). This problem can be overcome by changing the spring constant of the restraining springs of the instrument, such that less torque is required to turn the pointer by a given amount.

However, this reduces the ruggedness of the instrument and also demands better pivot design to reduce friction. This highlights a very important but tiresome principle in instrument design: any attempt to improve the performance of an instrument in one respect generally decreases the performance in some other aspect. This is an inescapable fact of life with passive instruments such as the type of voltmeter mentioned and is often the reason for the use of alternative active instruments such as digital voltmeters, where the inclusion of auxiliary power improves performance greatly.

Bridge circuits for measuring resistance values are a further example of the need for careful design of the measurement system. The impedance of the instrument measuring the bridge output voltage must be very large in comparison with the component resistances in the bridge circuit. Otherwise, the measuring instrument will load the circuit and draw current from it. This is discussed more fully in Section 9.

Example 1 Suppose that the components of the circuit shown in Fgr. 1a have the following values:

The voltage across AB is measured by a voltmeter whose internal resistance is 9500 Ohm. What is the measurement error caused by the resistance of the measuring instrument?

Solution:

Proceeding by applying Thevenin's theorem to find an equivalent circuit to that of Fgr. 1a of the form shown in Fgr. 1b, and substituting the given component values into the equation for RAB (1), we obtain:

Thus, the error in the measured value is 5%.

Errors due to Environmental Inputs

An environmental input is defined as an apparently real input to a measurement system that is actually caused by a change in the environmental conditions surrounding the measurement system. The fact that the static and dynamic characteristics specified for measuring instruments are only valid for particular environmental conditions (e.g., of temperature and pressure) has already been discussed at considerable length in Section 2. These specified conditions must be reproduced as closely as possible during calibration exercises because, away from the specified calibration conditions, the characteristics of measuring instruments vary to some extent and cause measurement errors. The magnitude of this environment-induced variation is quantified by the two constants known as sensitivity drift and zero drift, both of which are generally included in the published specifications for an instrument. Such variations of environmental conditions away from the calibration conditions are sometimes described as modifying inputs to the measurement system because they modify the output of the system. When such modifying inputs are present, it’s often difficult to determine how much of the output change in a measurement system is due to a change in the measured variable and how much is due to a change in environmental conditions. This is illustrated by the following example.

Suppose we are given a small closed box and told that it may contain either a mouse or a rat.

We’re also told that the box weighs 0.1 kg when empty. If we put the box onto a bathroom scale and observe a reading of 1.0 kg, this does not immediately tell us what is in the box because the reading may be due to one of three things:

(a) a 0.9 kg rat in the box (real input); (b) an empty box with a 0.9 kg bias on the scale due to a temperature change (environmental input);

(c) a 0.4 kg mouse in the box together with a 0.5 kg bias (real + environmental inputs)

Thus, the magnitude of any environmental input must be measured before the value of the measured quantity (the real input) can be determined from the output reading of an instrument.

In any general measurement situation, it’s very difficult to avoid environmental inputs, as it’s either impractical or impossible to control the environmental conditions surrounding the measurement system. System designers are therefore charged with the task of either reducing the susceptibility of measuring instruments to environmental inputs or, alternatively, quantifying the effects of environmental inputs and correcting for them in the instrument output reading. The techniques used to deal with environmental inputs and minimize their effects on the final output measurement follow a number of routes as discussed later.

Wear in Instrument Components

Systematic errors can frequently develop over a period of time because of wear in instrument components. Recalibration often provides a full solution to this problem.

Connecting Leads

In connecting together the components of a measurement system, a common source of error is the failure to take proper account of the resistance of connecting leads (or pipes in the case of pneumatically or hydraulically actuated measurement systems). For instance, in typical applications of a resistance thermometer, it’s common to find that the thermometer is separated from other parts of the measurement system by perhaps 100 meters. The resistance of such a length of 20-gauge copper wire is 7 Ohm, and there is a further complication that such wire has a temperature coefficient of 1 mOhm/ deg. C.

Therefore, careful consideration needs to be given to the choice of connecting leads. Not only should they be of adequate cross section so that their resistance is minimized, but they should be screened adequately if they are thought likely to be subject to electrical or magnetic fields that could otherwise cause induced noise. Where screening is thought essential, then the routing of cables also needs careful planning. In one application in the author's personal experience involving instrumentation of an electric-arc steelmaking furnace, screened signal-carrying cables between transducers on the arc furnace and a control room at the side of the furnace were initially corrupted by high-amplitude 50-Hz noise. However, by changing the route of the cables between the transducers and the control room, the magnitude of this induced noise was reduced by a factor of about ten.

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