Download 1 - Dr. Jonti Horner

Circuits 1: Direct Current Circuits
Sam Wann (and Anne Uther)
L1 Discovery Labs, Lab Group: A, Lab day: Wednesday
Submitted: 16/12/09 Experiment date: 18/11/09
Three different experimental techniques for determining the resistance of a resistor have been
investigated. All the techniques yield values that are consistent with each other and with the
manufacturer’s stated value of (100 ± 10) kΩ. The best value of the resistance, (99.7 ± 0.1) kΩ, is given
by the average of the values obtained from the three methods. It is found that an Ohmmeter is the best
method for making quick measurements, while taking multiple readings of current and potential
difference is the most precise method and the only one that verifies the Ohmic behaviour of the resistor.
1. Introduction
Resistors are used in electronic circuits to control the
potential difference passing through them. If a resistor is
too high then an electrical component will not work; if it
is too low then the increased potential difference through
the component may cause serious damage, for example,
bulbs blow and diodes fail. It is therefore very important
to measure the resistance of a component before use.
The resistance R (measured in Ohms, Ω) is given by
Ohm’s Law,1
R 
V
,
I
(1)
where V is the potential difference (p.d.) and I is the
current passing through the material.
In this paper, three different methods of determining the
resistance of a resistor labelled “100 kΩ” are presented.
This includes a single set of V and I measurements
(Method A), a series of I measurements for different
values of V (Method B) and a direct measurement of the
resistor with a specially made Ohmmeter (Method C).
After giving the results, we discuss how the values differ
and which method is most appropriate for various
experimental situations. Appendix I details how the
errors are calculated.
2. Experimental
A series circuit containing an unknown resistor from
Acme-Science Limited, a d.c. power supply and an
ammeter (a UNI-T multimeter) was set up as shown in
Fig. 1. A UNI-T multimeter was also used as a
voltmeter, added in parallel to the resistor. This circuit
was used for methods A and B.
For Method A, the power supply was set to a fixed p.d.
(~5 V) and the current through the resistor was
measured. Method B expanded upon this measurement
by varying the power supply between 0.5 V and 10.0 V
in increments of 0.5 V. The current was recorded for
each p.d. A linear fit was obtained using a least squares
fitting algorithm (LINEST in Excel) and the resistance
was determined from the gradient of the resulting line.
Fig. 1: Circuit diagram of the experimental set up used to
measure the resistance of an unknown resistor (R). The d.c.
supply (E) through the resistor was varied and handheld
multimeters were used for the ammeter (A) and voltmeter (V).
Method C required the use of a UNI-T multimeter as an
Ohmmeter. The Ohmmeter was connected to the resistor
and the resistance was measured directly.
3. Results
Table 1 shows the resistance of the resistor determined
by each of the three different methods. It can be seen
that the results all lie within the error of the others
meaning that they can be combined. The best value for
the resistor is 99.7 ± 0.1 kΩ. The error analysis
methodology is described in Appendix I.
Method
A (Single measurement)
B (Graphical method)
C (Direct measurement)
Combined measurement
Resistance (kΩ)
99.5 ± 0.3
99.82 ± 0.04
99.8 ± 0.1
99.7 ± 0.1
Table 1: Values of resistance and associated errors for each of
the three different methods. The error calculation methods are
described in Appendix I.
The results of Method B are shown graphically in Fig. 2.
The results show a linear relationship between the
current and the p.d. through the resistor. The gradient of
the graph was determined from a least squares fit and
from this the resistance and its error were determined.
Page 2
Sam Wann
component obeys Ohm’s Law. It is therefore deemed the
most suitable for quick measurements of components
that are known to be Ohmic, when the accuracy of the
resistance is not critically important.
120
Current (µA)
100
80
60
40
20
0
0
2
4
6
8
10
12
Potential Difference (V)
Fig. 2: The current through an unknown resistor as a function
of applied potential difference is plotted in order to verify
Ohm’s Law. A linear trendline and error bars (too small to be
seen) have been added.
Method B took the longest time to complete because it
required multiple measurements. Method C, the direct
method, required only a single measurement and needed
no further analysis.
Method A is the next quickest method, but it requires a
combination of two measurements, which increases the
uncertainty on the value. Like Method C, it does not
confirm whether the component being tested is Ohmic.
It is therefore best suited for situations where an
Ohmmeter is not available and speed is more important
than accuracy.
Using Method B increases the precision of the
measurements and has the additional advantage of
verifying that the resistor is indeed Ohmic. The
additional precision and information about the
component outweigh the only disadvantage, which is
that the technique takes a long time. This method is
therefore the most suitable when a high degree of
precision is required, and it is also the best for
investigating unknown electrical components.
4. Discussion
Ohm’s Law [Eq. (1)] states that there is a linear
relationship between the current and potential difference
through a resistor. The data collected via Method B and
presented in Fig. 2 demonstrate this linear relationship,
verifying that Ohm’s Law is applicable for this
component. Furthermore, the trendline in Fig. 2 passes
through the origin, indicating that there are no
significant systematic errors in the equipment used. This
verification confirms the validity of the results from
Methods A and B.
Method C uses an Ohmmeter, which also relies on
Ohm’s Law. The device passes a known current through
the resistor and measures the potential difference across
it. Since Method B has verified that the resistor is
Ohmic, this is a valid method for measuring the
resistance.
Table 1 shows that the values determined by all three
methods are consistent within their errors, confirming
that all the methods are suitable for measuring the
resistance of an Ohmic resistor. The results were
averaged to give the best value of (99.7 ± 0.1) kΩ. This
is consistent with the nominal value of 100 kΩ, given
that the manufacturer’s tolerance is ±10%.
The method used in an experiment depends upon the
information required by the experimentalist. Since all
the methods can be used successfully to give an accurate
value of resistance, it is important to evaluate the
advantages and disadvantages of each method.
Method C is by far the quickest method and has the
same uncertainty as the best value, 0.1 kΩ. However,
because it only uses one measurement and relies on the
accuracy of the Ohmmeter, it could be subject to large
random and systematic errors that would pass
undetected. This method does not determine whether the
5. Conclusions
The resistance of a nominal 100 kΩ resistor was
measured by three different methods. The values of the
resistance determined by each method were consistent
within their uncertainties. Averaging the results from all
three methods gives a best value of the resistance of
(99.7 ± 0.1) kΩ.
After discussing the three experimental methods, we
concluded that the choice of method depends on the
information that is required. For fast measurements of
the resistance, using an Ohmmeter is the best method; if
an Ohmmeter is not available, a single measurement of
current and potential difference is nearly as fast,
although less precise. If high precision and verification
of Ohm’s Law are required, the experimentalist should
take a series of current and potential difference
measurements and obtain the resistance from the
gradient of a linear fit to the data.
Acknowledgements: We thank Toby Broadhurst for
allowing us access to his data files.
References
[1] R. Wolfson, Essential University Physics
International Edition, Pearson Addison-Wesley: San
Francisco (2008).
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Sam Wann
Appendix I
1. Error analysis for resistance measurements
The values determined in this experiment all had an
degree of uncertainty. The error on the best value was
determined by combining the uncertainties of each
individual measurement.
Method A used two separate measurements to determine
the resistance RA via Ohm’s Law [Eq. (1)]. Calculating
the error αA involves combining the uncertainties of the
voltmeter (αV) and the ammeter (αI):
 V 

 V 
 A  RA 
2
 
 I 
 I 
2
.
(2)
.
The error αB on the resistance determined by Method B
(RB) was derived from the error on the gradient (αg) of
Fig. 2 using
 B  RB
g
g
,
(3)
where g is the value of the gradient determined from the
least squares fitting routine.
Method C used only one measurement to determine the
resistance RC. Therefore the error was taken to be equal
to the uncertainty of the digital Ohmmeter, αC = 0.1 kΩ.
2. Combining errors
The values of resistance found by the three different
methods were combined to find the mean resistance R
using the following equation:
R 
1
 RA  RB  RC  .
3
(4)
Each value of the resistance had a different error.
Therefore, the error on the mean resistance  R was
found from
R 
 R
RC
RB
A



   2    2    2
B
C
 A
 A 2
  B 
2
  B 




2
.
(5)