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m351 - MISN-0-351 EULERS METHOD FOR COUPLED DIFFERENTIAL...

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Project PHYSNET Physics Bldg. Michigan State University East Lansing, MI MISN-0-351 EULER’S METHOD FOR COUPLED DIFFERENTIAL EQUATIONS; RLC CIRCUITS q(t) q(t + t) D t t + ½ t D t + t D a b c 1 EULER’S METHOD FOR COUPLED DIFFERENTIAL EQUATIONS; RLC CIRCUITS by Robert Ehrlich 1. Euler’s Method a. Original Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 b. Improved Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 c. The RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 d. Applying the Two Euler Methods to RLC Circuits . . . . . . 4 2. Several Exact Solutions a. Zero Applied Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 b. Sinusoidal Applied Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 c. Resonance in RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 d. Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3. Computer Implementation a. Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 b. Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. A Project: Examine Accuracy a. Accuracy For Free Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 11 b. Accuracy For Forced Oscillations, Resonances . . . . . . . . . . 12 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A. Fortran, Basic, C++ Programs . . . . . . . . . . . . . . . . . . . . . . . . 12 2
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ID Sheet: MISN-0-351 Title: Euler’s Method for Coupled Differential Equations; RLC Circuits Author: R. Ehrlich, Physics Dept., George Mason University, Fairfax, VA 22030; (703)323-2303. Version: 3/11/2002 Evaluation: Stage 0 Length: 2 hr; 20 pages Input Skills : 1. Vocabulary: inductor (MISN-0-144). 2. Describe the Euler-method algorithm for solving first-order differ- ential equations (MISN-0-350). Output Skills (Knowledge) : K1. Write an explanation of the improved Euler method for the nu- merical solution of two coupled first-order differential equations of the form dy 1 /dx = f 1 ( y 1 , y 2 , x ), dy 2 /dx = f 2 ( y 1 , y 2 , x ). Output Skills (Project) : P1. Use the improved Euler method to solve the coupled, first order differential equations for an RLC circuit for free oscillations. P2. Use the improved Euler method to solve the coupled, first-order differential equations for an RLC circuit for oscillations that are forced by a sine-wave voltage generator. External Resources (Required) : 1. A computer with BASIC or FORTRAN or C++. 3 THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goal of our project is to assist a network of educators and scientists in transferring physics from one person to another. We support manuscript processing and distribution, along with communication and information systems. We also work with employers to identify basic scientific skills as well as physics topics that are needed in science and technology. A number of our publications are aimed at assisting users in acquiring such skills. Our publications are designed: (i) to be updated quickly in response to field tests and new scientific developments; (ii) to be used in both class- room and professional settings; (iii) to show the prerequisite dependen- cies existing among the various chunks of physics knowledge and skill, as a guide both to mental organization and to use of the materials; and (iv) to be adapted quickly to specific user needs ranging from single-skill instruction to complete custom textbooks. New authors, reviewers and field testers are welcome.
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