3.2 Sources of systematic error
Systematic errors in the output of
many instruments are due to factors inherent in the manufacture of the
instrument arising out of tolerances in the components of the instrument. They
can also arise due to wear in instrument components over a period of time. In
other cases, systematic errors are introduced either by the effect of envi[1]ronmental
disturbances or through the disturbance of the measured system by the act of
measurement. These various sources of systematic error, and ways in which the
magnitude of the errors can be reduced, are discussed below.
3.2.1 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. Whilst 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 analysing system disturbance
during measurements in electric circuits, Thevenin’s ´ theorem (see Appendix 3)
is often of great assistance. For instance, consider the circuit shown in
Figure 3.1(a) 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
E0 that existed prior to measurement. The extent of the disturbance can be
assessed by calculating the open[1]circuit voltage
E0 and comparing it with Em.
Thevenin’s theorem allows the circuit
of Figure 3.1(a) 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 Figure 3.1(b). 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 Figure 3.1(c). 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:
1 /RCD = 1 /(R1
+ R2) + 1 /R3 or RCD = (R1 + R2)R3
/(R1 + R2 + R3)
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:
1 /RAB = 1
/(RCD + R4) + 1 /R5 or RAB
= (R4 + RCD)R5 /R4 + RCD
+ R5
Substituting for RCD using the
expression derived previously, we obtain:
RAB = [{(R1 + R2)R3/ R1
+ R2 + R3} + R4] R5 /[{(R1 + R2)R3
/R1 + R2 + R3} + R4 + R5]
Defining I as the current flowing in
the circuit when the measuring instrument is connected to it, we can write:
I = E0 /(RAB + Rm) ,
and the voltage measured by the meter
is then given by:
Em = RmE0 /(RAB + Rm)
.
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:
Em /E0 = Rm /(RAB + Rm)
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.)
Example 3.1
Suppose that the components of the
circuit shown in Figure 3.1(a) have the following values:
The voltage across AB is measured by
a voltmeter whose internal resistance is 9500 . 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 ´ Figure 3.1(a) of the form
shown in Figure 3.1(b), and substituting the given component values into the
equation for RAB (3.1), we obtain:
From equation (3.2), we have:
Em /E0
= Rm /(RAB + Rm)
The measurement error is given by (E0
– Em):
E0 - Em = E0 { 1 - Rm /(RAB
+ Rm) }
Substituting in values:
E0 - Em = E0 {1 - 9500 /10 000} = 0.95E0
Thus, the error in the measured value
is 5%.
At this point, it is 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 (see Chapter 6 for further details). 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 greatly improves
performance.
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 resist[1]ances in the
bridge circuit. Otherwise, the measuring instrument will load the circuit and
draw current from it. This is discussed more fully in Chapter 7.
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