Lab 0: Modeling and Error. Goals 1. Review the lab procedures, safety issues and grading policies. 2. Understanding Modeling and Errors

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1 UNIVERSITY of PENNSYLVANIA DEPARTMENT of ELECTRICAL and SYSTEMS ENGINEERING Electrical Circuits and Systems II Laboratory ESE6 Introduction to the Lab Lab : Modeling and Error Goals 1. Review the lab procedures, safety issues and grading policies.. Understanding Modeling and Errors Pre-lab readings Before coming to the lab read the following (on the ESE6 webpage). 1. Lab policies and procedures. Musing about lab experiments 3. Introduction to this Lab (see below). 4. Bring your proto board and lab notebook to the lab. The lab notebook has to be a bound book, e.g. a "Lab Notebook" from Roaring Springs or "Computation Book" of Esselte. For details on the lab notebook see "Guidelines for lab notebook". Models are mathematical descriptions of the physical world. They fit nicely onto a sheet of paper, but they are always limited. You could probably describe yourself if a sentence, a paragraph, an essay, or even an autobiography if you had to. All are representations of yourself and none are wrong, regardless of their lengths. Mathematical models are the same. They can be simple, or they can be complex, but in the end, none are perfect, but none are completely wrong (unless of course, they are). We characterize this imperfection in error. Error is the deviation of the mathematical model from the physical reality, or depending on your perspective, the deviation of the physical reality from the mathematical model. One of the most challenging skills in science and engineering is figuring out how to make concrete statements about an imperfect system. For things which don t really matter, we can use words like around, near, or close to, but hopefully you will find yourself in a position someday in which your work does matter. In this case you ll need to not only make statements about what you do know, but also what you don t. 1

2 Understanding the measured data that characterizes the model, is of particular relevance for experiments such as those you will encounter in the lab. Introduction Descriptive Statistics Before we can learn how to manage imperfections, we must first learn how to talk about them. There are six key terms that we must know. Random Error: Random errors are errors that are not predictable. They can be characterized, and may be even explained, but they can not be anticipated. If enough measurements are taken and then averaged, the random error will disappear. Systematic Error: Systematic errors have a definite cause and show up in every measurement. May be the equipment needs calibrated or you are operating outside the specifications of your devices. The disadvantage is that you can t average them away, but the advantage is that even if their cause is not understood, they can be calibrated away. Precision: Precision is a description of the repeatability of a measurement, or in other words, the size of the random error. The measurement might be completely wrong, but so long as you get the same thing every time, the measurement would be considered precise. It seems silly at first that anyone would want an instrument which was consistently wrong, but a high degree of precision means that there is very little random error and what s left can either be calibrated or modeled away. Accuracy: Accuracy is a description of how close the measured result is to the true or desired outcome. In a sense it is a measure of the systematic error. Say you re firing a rocket to the moon. In this case, you would prefer a system which is accurate rather than precise since you only have one shot at it. Absolute: Absolute precision and accuracy has the units of measured quantity. In building a bridge machine all the pieces must fit together within fractions of an inch in order to operate smoothly regardless of their size. Relative: Relative precision and accuracy is unitless and is a measure of the error relative to the magnitude of the pertinent quantity. For instance getting a result of 13 when the true value was 14 is very impressive, but not so impressive if the true value was 5. These concepts pervade all of science and engineering and they will serve you well if you can fluidly move between them. Now we will provide mathematical definitions which are relevant to them. Note that the definition of accuracy and precision differ depending on who you talk to, but these will work for us. Suppose you have a data set X with N members which were expected to take the value of x. ref

3 x x max min ( = x = Max = ) x = Min mean = x = ( X) ( X) ( i ) ( ) ( ) ( ) ( x x) ( x ) i Standard Deviation = σ X = N = N x Precision of xi Rel. Precision of x = Abs x x = Abs x x x i N Accuracy of x = Abs x x i i ref i x i i Rel. Accuracy of xi = Abs xi xref / xref Check that the mathematical and conceptual definitions are congruent for you. Uncertain Numbers Uncertainty can be expressed in a variety of ways. The meaning is usually subtly embedded in how a number is written. The statement that x is equal to the number fifty 4 thousand can be expressed as x = 5, x = 5.k, x = 5k( ± 1%), x = 5. 1, or even x = 5( 1). All of these statements are saying the same thing with different levels of assuredness. Let s take a moment and figure out just how confident these numbers are. x = 5. 5 is a pure number. It is saying that x is equal to exactly fifty-thousand, no more, no less. This is a very powerful statement when applied to a continuous number. You might have exactly five-thousand eggs, or people, but you will never own a 5 Ohm resister. Other pure numbers include π, e, and imaginary i. These numbers have no uncertainty in your calculations. x = 5.k. The inclusion of the decimal point here indicates a degree of precision. It is an implicit promise that x is within the set [4995, 55]. x = 5k( ± 1%). By itself, 5k carries the same weighty meaning as 5, however, we degrade the statement by adding the ( ± 1%) and so only guarantee that x is within the set [45, 55]. Using percent error notation, we can specify uncertainties impossible using only decimal point positioning. 4 x = This example illustrates the beauty of scientific notation. It is a guarantee that x falls within the set [45, 55]. Note that there is no way to write this number using the k. For instance, 5. k is too strong of a promise for what we re trying to say. x = 5( 1). This is a common notation in books of physical constants and is very convenient. It implies that x falls within the set [49988, 51]. 3

4 Error Propagation A foot soldier was sent out to scout an enemy castle. Upon returning to the king he declared that there were 117 enemy soldiers there. The king was amazed that his man was able to get such an exact number and inquired how he did it. To this, the soldier replied I knew there were about a thousand in the castle and I counted seventeen standing guard on the walls. Errors propagate through your calculations. Not only do they propagate, but they also accumulate. The problem isn t that the soldier stated too many digits. It is probably true that the enemy numbers more than a thousand. The problem is that he didn t say what he doesn t know. We will now discuss several methods of doing calculations with uncertain numbers. Interval Arithmetic Interval arithmetic is the only exact and most difficult method of dealing with uncertain numbers with finite bounds. It requires good mathematical intuition. It is just coming into its own as a branch of mathematics, motivated by the need for programs which are guaranteed to work. (Floating-point numbers used in computers are ultimately approximations just as 5k Ω resister is only approximately 5k Ω.) Interval arithmetic is the only way to propagate uncertain numbers through a calculation and not use any approximations. The idea is very intuitive, although implementation can be a bit of a trick. Every operation between one or more intervals will yield a new interval which ranges from the minimum to the maximum possible value. Consider the following unary and binary operations. Graphically they can be shown as where the blue lines represent the input inverval(s), while the red lines represent the output intervals, while the plots show the functions which relate them. Notice how the bounds on the red interval does not necessarily having anything to do with the evaluation of the endpoints. 4

5 Let us do a couple simple numerical examples. Suppose A = 3. 1, B =. 1, and C = 3. 1 [.95.5] [ 1.5.5] [ ] [ ] 5( 11% ) A+ B = + = + + = = ± [.95.5] [ 1.5.5] [ ] [ ] 1.66( 7% ) AB= = = = ± Notice that in the end, we aren t sure of any digits and were forced to use percent error to express the number. We were barely sure of the in B, and so when combined with the uncertainty in A, we can t specify any digit for certain. These examples were easy. Lets look at some harder ones. Suppose A = 1. 1 and B = Note that they are independent variables even though they happen to be the same thing. In this case A B = [.1.1] ( A B) = [.1] [.1.1] 1( A B) = [ 1 1] sin ( 1( A B) ) = [ 1 1] [ ] A very interesting article on interval arithmetic can be found in the references. Gaussian Error Propagation The Gaussian (or normal ) distribution is the most used distribution in science and engineering. When using a Gaussian distribution, there exists a finite probability of obtaining a number within any interval, positive or negative. It follows therefore that even for simple relations, any result is theoretically possible. However, some results are much more likely than others. In this sense, an operation between two gaussian distributions will not yield a guarantee like an operation between two intervals. However, when the input distributions are contained within a limited enough range, much can be stated with reasonable assuredness about the output distribution. Since a Gaussian distribution makes no guarantee on the bounds of a number at all, interval arithmetic does not make sense here. Instead we must propagate the Gaussian distribution through a calculation based on their standard deviations and means. 5

6 The idea is relatively simple. Most functions are relatively linear over a small enough range. Therefore the standard deviation of the result in proportional to the standard deviation of input and can be imagined as a reflection of the input distribution across a line drawn tangent to the function Suppose we have three variables with means of X, A, and B, each with an uncertainty characterized by σ X, σ A, and σ B respectively and means of x, a, and b respectively such that X is a function of A and B If the uncertainties are small enough, σ x x a σ = + b σ X A B. Monte-Carlo Analysis Monte-Carlo is a gambling city in Monaco. Monte-Carlo analysis is a numerical technique in which each variable is represented by a data set. For instance, if A could have any value between al and ah with equal probability, then A could be represented by random number generated in MS Excel by ( a a) + a) ( ) RAND ( ) Likewise if B is a Gaussian distribution centered on b with a standard deviation σ B then = NORMSINV ( RAND ()) σ + B b again in the parlance of MS Excel. 6

7 Suppose obtain a new number X = A+ B. Every time we calculate this number, we will obtain a new value. If we were to perform the calculation many times, we would have a very good idea of the uncertainty in X given the uncertainty in A and B. We could also see how the error distributes itself, even though we are dealing with two different distributions. For formulas with many variables, different random distributions, and nonlinear relations, this is a very effective way of determining just what your answer will look like. Prelab: (answers to be handed in with the lab report at the start of the next lab) 1. Which will be greater, your accuracy or your precision? Why?. Which would you rather have, an instrument which is accurate or precise? How does your answer depend on your ability to calibrate the instrument? 3. Which would you rather have, systematic errors or random errors? How does your answer depend on the number of measurements and your ability to model your system? Let R 1= 1. 1 ( ± 5% ) Ω while R 1=. 1 ( ± 5% ) Ω. If you build a voltage divider such that Vout = Vin ( R ( R1+ R) ). Over what range can you guarantee that Vout V in will occupy? Use interval arithmetic. 5. Suppose you have two experimental quantities, A and B, with a mean of a and b and small standard deviations of σ A and σ B, respectively. Using A, B, and C find σ C if C= A+ B. find σ C if C= A B. find σ C if C = AB. find σ C if C= A B. find σ C if C= B ( A+ B ) 6. Build a Monte-Carlo simulation in MS Excel where you determine Vout/Vin if Vout=Vin (R/(R1+R)) and R1 and R have means of Ohms and. 1 3 Ohms respectively with standard deviations of. Ohms and 4. Ohms respectively. Lab Experiment Equipment: 1. HP multimeter HP 3441A. HP function generator/waveform generator (HP 331A) 3. HP digital oscilloscope HP kΩ resistors. 7

8 5. kω resistors kΩ resistors 7. PC with Excel spreadsheet Explore the Random and Systematic Errors in Resistors 1. Measure 1k resistors using HP 3441A. Use the maximum number of digits the multimeter will allow. Avoid actually touching the resistors directly with your fingers. Your fingers will warm the resistor and it will take a while for the resistor to return to room temperature. Record the results in MS Excel and your lab notebook. What is the mean and the standard deviation of your results? (Hint: Use =AVERAGE( _ : _ ) and =STDEVP( _ : _ ) ) What is the average precision (absolute and relative) of the resistors? What is the average accuracy (absolute and relative) of the resistors? What is the minimum and maximum value? (Hint: Use =MAX( _ : _ ) and MIN( _ : _ ) ) Create a scatter plot of the values. What does the distribution look like? The right way to do this would be to use a histogram, but this is difficult to do with so few measurements.. Repeat the same steps for k resistors. 3. Select a 1k and a k resistor at random. Build a voltage divider to yield that V 13V. Use 1V as V input. Measure V using the 3441A. Replace the out ( ) in in resistors into your stash and grab more at random and repeat the measurement. Do this for a total of times. Does the mean of these measurements make sense? Does the standard deviation make sense according to Gaussian error propagation? Does any value fall outside the range you calculated in the prelab using interval analysis? Assuming that 1k and k are the right values, discuss how the systematic and random errors have propagated through your system. out Modeling of Systematic Errors Random errors have the advantage that if you make a measurement enough times, the randomness should cancel out to yield the correct value. Systematic errors have the advantage that they can be characterized and removed. Measure five 56k resistors. What is the mean and standard deviation? What is the max and minimum value? Create the following simple circuit with the 56k resistors. Use alligator or mini clips rather than probes. 8

9 What is the Vpp across R? Is this value within the bounds you expect? Is the shape what you would expect? (hint: it shouldn t be) You have just stumbled across a systematic error. Every device has its own limitations. The goal of a manufacturer is not to remove them, but rather to make them as unobtrusive as possible. One of the best ways to do that is to state them clearly in such a way as they can be easily taken into consideration. What does it say between the probe inputs of the oscilloscope? Consider this alternative model where Rs=5Ohm, Rp=1MOhm, and Cp=13pF. Measure R1 and R using the multimeter. Calculate what Rp is based on your measurement of Vpp across R. Is it close to 1 MOhm? (Hint: you can ignore Rs if you like) Place another 56k resistor in parallel with the original R to yield a new R which is half the original value. Use your calibrated system to measure the new R. Is Vpp across R what you would expect now? In other words, have you successfully modeled out the systematic error? To hand-in Write your lab report, according to the guidelines, summarizing the experiments and results obtained in the lab. Include also the answers to the pre-lab questions. 9

10 The lab report is an important part of the laboratory. Write it carefully, be clear and well organized. It is the only way to convey that you did a great job in the lab. The typewritten report is due 48 hours after the end of this lab. Please, consult the Grading Policies at: References for further reading 1. Interval Arithmetic: 3_37.pdf. Gaussian Error Propagation: 3. Monte Carlo Simulations: ml 5. Brian Edwards January 11, 7 1