## EP-1 ERROR PROPAGATION SIMULATION
This experiment provides an example of (1) a distribution of random values, (2) how to express the dispersion of such a distribution (a measure of "error" or "uncertainty"), and (3) how the error measures of two variables relate to the error measure of a mathematical combination of those variables.
A number of sets of ten loose resistors of the same marked value in a numbered envelope (or box). A multimeter with ohmmeter function.
This is not a laboratory "experiment" in the usual sense, but a learning exercise to demonstrate the mathematical rules of error analysis. You will study a situation which simulates error distributions and error propagation Many manufactured items, produced by automated machinery or an assembly line, show a natural variation in some physical property: size, color, etc. This variability is a natural consequence of the many individual "errors" in each part of the fabrication machinery. Carbon resistors, used in all kinds of electrical and electronic circuits, are a good
example. They consist of a small chunk of carbon with two wires attached, then encased in
molded plastic for durability. Their electrical If you have a large bin of resistors, all manufactured to have the "same" resistance, they will, if measured precisely enough, have different resistance values. We will use this variability of resistance as a simulation of the variability we find in repeated measurements taken in a laboratory investigation. This variability of resistance values will be studied as a simulation of the indeterminate
error one observes in repeated 4. PRE-LAB PREPARATION: Before coming to laboratory, read the handout sheets which discuss experimental errors (uncertainties). In this experiment we are only concerned with Do the following exercises by using the rules given in the handout. [The starred questions are designed to separate the "A" students from the rest!]
(1) The
R = X + Y + Z + .... where X, Y, Z, etc. are the resistances of the individual resistors and R is the resistance of their series combination. Work out the error propagation equation for the formula for this law. Express it in two forms, one for absolute error and one for relative (fractional) error.
(2) The
1 1 1 1 — = — + — + — + ... R X Y Z Work out the error equation for the formula for an arbitrary number of resistors combined in parallel. Express it in both forms also. Hint: Since the numerator value "one" has no error, the relative (fractional) error in 1/X is, by the quotient rule,
Each term of the parallel resistor equation is of this form, so start by writing its error
equation in relative error form. Then algebraically rearrange and simplify it. [Answer: The
relative error in 1/X is ΔX/X (3) Two resistors are chosen from a batch in which all resistors are marked with the value 15 kilohms and a error limit of 10%. When they are connected in series, their combined resistance is expected to be 30 kilohms. What is the error limit of this result? Answer: 0.30k, which is 10% of 30k. (4) Two 10% resistors of size 15 kilohms are chosen from the same batch are connected in parallel. Their combined resistance is expected to be 7.5 kilohms. What is the error limit of this result? Answer: 0.75k, which is 10% of 7.5k. (5) Were you surprised at these results? Do not be misled by these "special" cases. The
results are Answer: 5k, which is 16.7% of 30k. Notice what this answer is (6*) Consider a large number of resistors of many different sizes all having 10%
uncertainty. Show algebraically that when (7*) Show the same for a parallel combination. (8*) A more realistic calculation takes into account the tendency of indeterminate errors in combinations of independent things have errors less than the maximum-error `worst-case' calculations above. The rules above are modified by `summing in quadrature' instead of summing. That is, the error equation in standard form is modified by squaring each term. Work out the results for the error in series and parallel combinations of two equal resistors and compare with the results above. [Answers: Series: 1.414 instead of 2. Parallel: 0.354 instead of 0.5.]
Use a digital multimeter as an ohmmeter. Its Do When finished with the experiment be sure to return the multimeter's range switch to the
Carbon resistors have been placed in envelopes (or boxes), 10 resistors to a box. These are numbered, and will be called "sets" from here on. In all parts of this experiment you will, at any time, work with just one of the numbered
sets of 10 resistors. Keep in mind that this exercise simulates experimental error in Since we wish to simulate All of the sets taken together represent a a sample from a larger This experiment is a concrete simulation model of the measurement process. You will
make a finite number of measurements of a quantity in order to draw generalizations about
the much larger set of measurements which you could make (but won't), to specify how well
the small sample represents the larger set. [The set of (1) Use the digital multimeter. Set its range switch to the ohmmeter function. Use the
ohmmeter to measure each of the resistors in one set. To assure unbiased, random selection
in later parts of the experiment, do (2) When finished with the ten resistors of one set, trade sets with someone else who has finished. You will trade envelopes with other students later, but be sure never to repeat measurements on a set you have measured before! Each new set of resistors represents a fresh random sample from the much larger sample of over 120 resistors available to you. (3) Find the mean of 20 resistance values (two sets), the maximum deviation, the average deviation of the mean, and [if you wish] the standard deviation of the mean. (4) Obtain another envelope. Group the ten resistors in 5 sets of pairs. Use the ohmmeter to measure the resistance of each pair of two connected in series (5 measurements). Obtain another envelope and repeat, until you have a total of twenty resistance values of series pairs. [You should be pretty efficient at this by now!] (5) Obtain a new set of resistors. Repeat the procedures (3) and (4) again, but this time connect resistors in parallel, for a total of twenty measurements. (6) If there it time, investigate the situation of
(1) For each group of 10 resistance values, find the mean, maximum deviation, and the average deviation of the mean. [The average deviation of the mean will be taken as the "uncertainty" of the mean.] (2) For each group of 20 measurements of two resistors in series, do the same. (3) For each group of 20 measurements of two resistors in parallel, do the same.
The following questions assume that you are (1) The error equation you derived for series resistors predicts that the absolute uncertainty in the resistance of the series combination is the sum of the absolute certainties of the individual resistors. Does your data support this? If not, would you say that you have invalidated the rule? (2) The error equation you derived for parallel resistors predicts that the relative uncertainty in the resistance of the parallel combination is the sum of the relative uncertainties in the individual resistors. Does your data support this? If not, would you say that you have invalidated the rule? (3) Are the results for series and parallel resistor combinations in sufficiently good agreement with the error-propagation rules for maximum uncertainties (error limits)? For average deviations of the mean? Discuss. (4) If you could have measured all of the resistors the factory manufactured, you could
make some very definite statements about the average value and the distribution of the measured
values. But you have only a small subset sample of this larger "parent" distribution. Still you
can draw some conclusions about the parent distribution on the basis of your sample. Is the
parent distribution symmetric? Is it Gaussian? If it isn't Gaussian, how would you describe it?
[Hint: Plot a distribution curve of (5) You worked with 20 values in each part of the experiment. Was this enough to
demonstrate the laws of error propagation? Would 50 have demonstrated the laws better?
If less time were available, and you had only used 5 values, would that have been
enough to demonstrate the same principles and results convincingly? Answer this on the basis
of © 1995, 2004 by Donald E. Simanek. |