This article was downloaded by: [RMIT University] On: 02 September 2013, At: 12:18 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/upri20 A LABORATORY EXPERIENCE FOR STUDENTS OF DIFFERENTIAL EQUATIONS USING RLC CIRCUITS Jeff Graham PhD a & Julia Barnes PhD a a Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC, 28723-9049, USA Published online: 13 Aug 2007. To cite this article: Jeff Graham PhD & Julia Barnes PhD (1997) A LABORATORY EXPERIENCE FOR STUDENTS OF DIFFERENTIAL EQUATIONS USING RLC CIRCUITS, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 7:4, 334-340, DOI: 10.1080/10511979708965875 To link to this article: http://dx.doi.org/10.1080/10511979708965875 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any
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This article was downloaded by: [RMIT University]On: 02 September 2013, At: 12:18Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T3JH, UK
PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20
A LABORATORY EXPERIENCEFOR STUDENTS OFDIFFERENTIAL EQUATIONSUSING RLC CIRCUITSJeff Graham PhD a & Julia Barnes PhD aa Department of Mathematics and ComputerScience, Western Carolina University, Cullowhee,NC, 28723-9049, USAPublished online: 13 Aug 2007.
To cite this article: Jeff Graham PhD & Julia Barnes PhD (1997) A LABORATORYEXPERIENCE FOR STUDENTS OF DIFFERENTIAL EQUATIONS USING RLC CIRCUITS,PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 7:4,334-340, DOI: 10.1080/10511979708965875
To link to this article: http://dx.doi.org/10.1080/10511979708965875
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for any
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ADDRESS: Department of Mathematics and Computer Science, WesternCarolina University, Cullowhee NC 28723-9049 USA.
ABSTRACT: Although differential equations are billed as applied mathematics, there is rarely any hands-on experience incorporated into thecourse. In this paper, we present a laboratory project that requiresthe students to obtain data from a physics lab and use that datato compute the coefficients of the second order differential equationwhich mathematically models the behavior of an RLC circuit.
Due to the recent calculus reform movement, there is growing interest inhow differential equations is taught to undergraduates. Most instructorsagree that the former method was too much of a "cookbook approach";however, there is no clear consensus as to what should be done instead.At our institution, we do not have an engineering program. Most of ourdifferential equations students are math and science majors. Therefore, wedecided to incorporate a hands-on science laboratory component into thecourse. The rationale for including such a component is that the labs assistin bringing mathematics alive to the students. Presented in this paper isone of the lab assignments.
A typical problem from a differential equations text is to solve the following second order equation for voltage, v(t):
LCv"(t) + RCv'(t) + v(t) = f(t)
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given R,L,C,!(t) , and some initial conditions. The problem posed in thisway is called a forward problem. A well-posed forward problem has threenice mathematical properties. The problem has a solution, the solution isunique, and the solution depends continuously on the parameters in theequation.
In this project, students are given an RLC circuit with unknown resistance R, capacitance C, and inductance L. Students are asked to determinethe parameter values RC and LC found in Equation 0 by taking measurements of the solution to the forward problem. This type of problem is calledan inverse problem. Inverse problems typically violate at least one of thethree conditions necessary for being well-posed.
We can think of an inverse problem as trying to determine the appropriate forward problem. A familiar example of an inverse problem is determining the anti-derivative of a function. We all know that the anti-derivativeproblem violates the uniqueness property. This lab violates the uniquenessproperty as well in that R, L, and C cannot be uniquely determined.
For a more detailed discussion of inverse problems see [2]. It has a largeannotated bibliography which is a great source of ideas.
2 TEACHER'S NOTES
This lab does require the teacher to set up a circuit before the lab begins.Since we used equipment from a physics laboratory, we contacted one ofthe physics professors at our university for assistance. 1 We explained thatwe were interested in having students recover data about the differentialequation associated with an RLC circuit system, and we were met withenthusiasm. He provided lab space, a circuit board, a collection of resistors,conductors, and capacitors, 2-channel oscilloscope, and a function generator.He also showed us how to use the equipment. Figure 1 shows the schematicof the circuit we used.
In class, we discussed the physics behind RLC circuits, and worked withsome textbook examples of second order differential equations associatedwith these systems. Then we gave a brief overview of the inverse problemthat we intended to solve in the lab.
We only had enough equipment for one group to work at a time, so eachlab group signed up for a 15 minute time block in the physics lab. Wemet the students there and showed them how to use the equipment. They
1 We could not have done this lab without cooperation from the physics department.We would especially like to thank Kurt Vandervoort for his support and willingness toprovide us with equipment.
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gathered all the necessary data (described more fully in Section 3.3), andfinished working on the questions away from the lab. Between lab groups,we switched the capacitor in order to give each group different data.
R
F(t)
Gnd
Figure 1. The Circuit Diagram.
L
Since equipment may vary from physics department to physics department, we suggest that anyone planning on to do this lab consult their localphysicist - it is not only helpful, but it improves interdisciplinary relationships. One could also use a software package such as SPICE to simulate thecircuit. Since hands-on experiences in mathematics are lacking, we preferto use the physical devices.
Throughout the lab, students are presented with questions to help themanalyze the system. Many of the questions are straight-forward, but somerequire a little more thought. For example, number 8 in Section 3.3 asksstudents if obtaining more data would enable them to solve explicitly forR, L, and C. Students generally feel that having more data will make itpossible to separate the values of R, L, and C, but in actuality, more datadoes not help. More data would allow them to use regression to obtainbetter estimates for RC and LC, but it will never separate C from theother variables. However, it may be possible to determine the parametersR, L, and C uniquely if you choose to measure current instead of voltage.See [1] for ideas on how this might be accomplished. Another questionwhich is difficult for students is number 3 in Section 3.2. Here we ask whyit is possible to ignore the solution to the homogeneous equation. Since RCis typically much larger that LC, the roots to the characteristic equationare negative, so the homogeneous solution decays rapidly enough that themain effect comes from the particular solution.
We include our handout for students in the following section.
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3 RLC LAB HANDOUT
In this lab you will recover the coefficients LC and RC in the 2nd orderODE
LCv"(t) + RCv'(t) + v(t) = J(t) (1)
using measurements of the voltage v(t) . This handout contains the background information necessary to perform this task. Read through this information and answer as many questions as possible BEFORE going tothe physics lab. Include answers to all of the questions in your lab report,as well as all of your results.
3.1 METHOD
Two different forcing functions of the form f;(t) = F;COS(Wit) are usedand the amplitude of the resulting steady state is measured. The analyticsolution should have amplitude
A-- Fi ()• - "/(1- LCwl)2 + R2C2W [ 2
where i = 1 or 2. For simplicity in the lab, we use the same amplitude, Fo,for both forcing functions. Using elementary algebra it is possible to solvefor x = LC and y = RC, the coefficients in Equation 1.
3.2 BEFORE GOING TO THE LAB...
Answer the following questions.
1. Suppose that Ji(t) = FoCOS(Wit) and the amplitudes of the corresponding steady state solutions, Al and A2 , are known. Solve thesystem for x = LC and y = RC.
2. Where did Ai come from in Equation 3?
3. The solution to a non -homogeneous differential equation is v(t) =Vh(t) + vp(t), where Vh(t) is the general solution to the correspondinghomogeneous equation, and vp is the particular solution. In findingthe solution to Equation 2 we are ignoring the effect of Vh. Why?
4. The differential equation in Section 5.4 [2, p. 218) is
Lq"(t) + Rq'(t) + q(t) = E(t).C
How is this equation related to the one we are using?
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3.3 Gathering and using the data
Go to the physics lab. An RLC circuit has been built for your convenience.It is hooked to the oscilloscope on two channels. Channel one has thesinusoidal input (the forcing functions) . The frequency is adjustable anddisplayed on the function generator to your right. If we display channel oneon the scope, we can measure the amplitude of the input signal. The bestway to measure the amplitude is to adjust the scale to something convenient,determine the vertical distance between the maximum and minimum of thecurve, and divide by two. If we switch to channel 2, we can do the samething for the steady state amplitudes (i.e. the amplitude going out). It iswise to set the amplitude going in to be the same for both frequencies.
1. Find the resonant angular frequency of this circuit. Describe how youknow this is the resonant angular frequency.
2. Fill in the following table of data for a fixed input amplitude and twodifferent frequencies. Do not use the resonant angular frequency. Wewill be calculating the resonant angular frequency from this data andcomparing it to your answer in number 1.
Frequency Amplitude In Amplitude Out
3. What happens to the amplitude going out when the frequency movesaway from the resonant frequency?
4. In order to solve for LC and RC, is it important to have measurementsfor at least 2 different frequencies? Would 1 be enough? Why or whynot?
5. Evaluate the values of LC and RC using the equation you solvedbefore going to the lab. Rewrite the original differential equation(Equation 2) using your values of LC and RC.
6. If you select different frequencies, should you get different values forLC and RC? Why or why not?
7. Calculate the value of the resonant angular frequency (,,. in your book[2, p. 214]). Compare this to your answer in question 1. Is this answerreasonable?
8. If you add another line of data in the chart above, would you be ableto solve specifically for L, R, and C? Why or why not? (Think aboutthis before answering.)
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9. Suppose that C = 50 microfarads. Determine the values of Land R.
10. If you wanted to have a spring/mass system (a spring hanging fromthe ceiling with a mass attached to it) that satisfies the differentialequation you have solved for in question 9, what would be the weightof the spring, the spring constant, and the damping coefficient?
11. Write a conclusion to this lab. You should include a brief summary ofwhat you were trying to solve as well as how this could generalize toother RLC circuits. Also, include any situations that you think maybe solved using the methods from this lab .
4 STUDENT REACTIONS
This and other labs during the semester seemed to be well received by thestudents. Besides being a nice break from doing straight computations andgiving students the opportunity to work in groups, the students felt that ithelped them understand the material better. Here are some comments thatstudents made on course evaluations:
• When I did homework problems, I didn't completely understand whatwas going on. The group projects helped me understand some of thephysics behind the problems and cleared up many questions aboutdifferential equations.
• I enjoyed the labs. They helped bring the math into real world applications.
• Labs were neat and interesting. They were fun to do.
One other comment from students had to do with the title of the lab.They thought that "RLC Lab" sounded uninteresting. Maybe we shouldcall it "An Electrifying Experience" next time.
REFERENCES
1. Graham, Jeff. 1995. Using Inverse Problems as Projects in a Differential Equations Course. Electronic Proceedings of the Eighth AnnualConference on Technology in Collegiate Mathematics. Online. Available:http://archives.math.utk.edu/ICTCM/EP-8.html.
2. Groetsch, Charles W. 1993. Inverse Problems in the MathematicalSciences. Weisbaden GERMANY: Vieweg.
3. Nagle, R. Kent and Edward B. Saff. 1993. Fundamentals Of Differential Equations, Third Edition. Reading MA: Addison Wesley.
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BIOGRAPHICAL SKETCHES
Julia Barnes and Jeff Graham are both Assistant Professors at WesternCarolina University in Cullowhee NC. Julia Barnes received her PhD inmathematics from the University of North Carolina at Chapel Hill in 1996,and does research in the areas of complex dynamical systems and ergodictheory. Jeff Graham received his PhD in mathematics from RensselaerPolytechnic Institute and does research in the areas of scientific computingand numerical analysis.
