EP-1 ERROR PROPAGATION SIMULATION

1. PURPOSE

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.

2. APPARATUS

A number of sets of ten loose resistors of the same marked value in a numbered envelope (or box).

A multimeter with ohmmeter function.

3. DESCRIPTION OF THE INVESTIGATION

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 resistance is their important property, and that resistance shows variation from resistor to resistor.

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 independent measurements of any physical quantity. Normally the experimental error in a measurement is due mostly to instrumental and procedural variations, But in this simulation the variation is entirely in the things being measured. Instrumental error is negligible by comparison, and will be ignored.

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 indeterminate errors. Therefore the rules given in Appendix II of the handout. We will not ask you to calculate or use standard deviations, though serious students might want to do this for practice.

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!]

Fig. 1. Resistors connected in series.

(1) The physical law for the effective resistance of a series string of resistors is:

[1]

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.

Fig. 2. Resistors connected in parallel.

(2) The physical law for the effective resistance of resistors combined in parallel is:

[2]

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,

Relative error Relative error Relative error in numerator - in denominator = in quotient.

ΔX ΔX 0 - —— = - —— X X

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/X².]

(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 not this simple when the two resistors have different uncertainty values. Consider one 20% resistor of size 20k and one 10% resistor of size 10k connected in series. The combined resistance is therefore 30k. What is the expected % error in this result?

Answer: 5k, which is 16.7% of 30k.

Notice what this answer is not. 16.7% is not the sum of 20% and 10%, and is not the average of 20% and 10%. This should illustrate that you must not jump to hasty conclusions based on "common sense." Work out these problems mathematically!

(6*) Consider a large number of resistors of many different sizes all having 10% uncertainty. Show algebraically that when any number of these resistors are connected in series, the resistance of the combination will also have 10% uncertainty.

(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.]

5. USE OF THE APPARATUS:

Use a digital multimeter as an ohmmeter. Its range switch has a setting for resistance measurements. Its two probes are then connected to the lead wires of the resistor (or combination of resistors) and the resistance value appears on the digital display.

Do not bend or twist the resistor leads, leave them straight. If you wish to make firmer connections, use alligator clips to attach to the resistor leads.

When finished with the experiment be sure to return the multimeter's range switch to the volts position, and switch the meter off. The prevents the possibility of the meter's internal batteries being drained.

6. PROCEDURE:

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. Do not mix sets or interchange resistors between sets. Have only one set "open" on your lab table at a time.

Keep in mind that this exercise simulates experimental error in any measurement process. It is instructive to think of each set of resistors as a set of ten measurements you might make of any physical quantity.

Since we wish to simulate independent measurements, it is important that you never measure the same resistor more than once.

All of the sets taken together represent a a sample from a larger set of sets of measurements: the entire batch of resistors of this value which the factory manufactured.

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 all measurements which could possibly be made is called the parent distribution.]

(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 not mark or label the individual resistors, but return each to the envelope after it has been measured (to avoid measuring the same one twice). Of course you will record the ten values you obtain, and you must record the number on each set's envelope (to prevent you from using the same envelope twice).

(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 three randomly selected resistors in parallel. You might also investigate two in parallel connected to another in series.

7. ANALYSIS

(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.

8. DISCUSSION QUESTIONS

The following questions assume that you are not using standard deviations as the error measures.

(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 all of the resistors you measured.]

(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 your data.