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. Even as 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 is an important consideration in instrument design. However, an accurate understanding of the mechanisms of system disturbance is a prerequisite for this.

Measurements in Electric Circuits

In analyzing system disturbance during measurements in electric circuits, Th´evenin’s theorem is often of great assistance. For instance, consider a circuit in which the voltage across resistor R5 is to be measured by a voltmeter with resistance Rm. 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 E0 that existed prior to measurement. The extent of the disturbance can be assessed by calculating the open circuit voltage E0 and comparing it with Em. Th´evenin’s theorem allows the circuit comprising two voltage sources and five resistors to be replaced by an equivalent circuit containing a single resistance and one voltage source. For the purpose of defining the equivalent single resistance of a circuit by Th´evenin’s theorem, all voltage sources are represented just by their internal resistance, which can be approximated to zero. 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:

[pmath]1/RCD=1/(R1+R2)+1/R3[/pmath]    OR  [pmath]RCD=((R1+R2)R3)/(R1+R2+R2)[/pmath]

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

Defining ‘I’ as the current flowing in the circuit when the measuring instrument is connected to it, we can write:

[pmath]I =E0/(Rab+Rm)[/pmath]

and the voltage measured by the meter is then given by:

[pmath]Em=RmEo/(Rab+Rm)[/pmath]

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 E0. The effect of measurement is therefore to reduce the voltage across AB by the ratio given by:

[pmath]Em/Eo=Rm/(Rab+Rm)[/pmath]

It is thus obvious that as Rm gets larger, the ratio Em/E0 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 E0, since this is not required in quantifying the effect of Rm.)

R1= 400Ω ; R2= 600Ω ; R3=1000Ω , R4=500Ω ; R5 =1000Ω

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.