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Unformatted text preview: Math 216: Differential Equations Lab 2: Euler’s Method and RC Circuits Goals In this lab you will implement Euler’s method to approximate measurements of the charge on a capacitor in a basic RC circuit. You will learn how to write .m files for Matlab and how to program Euler’s method; then you will investigate some of the limitations of the method. Application: a basic RC circuit The state of an electrical circuit consisting of resistors, capacitors, and an applied voltage can be described by differential equations. Consider the following circuit R S C I E(t) +- with resistance R Ohms (Ω), capacitance C Farads (F), and applied voltage E ( t ) Volts (V). The charge on the top plate of the capacitor at time t is Q ( t ) Coulombs (C), and the current through the resistor is I ( t ) Amperes (A). The resistor has a voltage drop of RI , and the capacitor has a voltage drop of Q/C . When switch S is closed at time t = 0, the sum of the voltage across the resistor and the capacitor must equal the applied voltage. This gives us the equation RI + 1 C Q = E ( t ) . The current in the circuit is the rate of change of the amount of charge on the capacitor. So using the relationship dQ/dt = I , this becomes a first order differential equation for Q ( t ): R dQ dt + 1 C Q = E ( t ) . The initial condition for this equation is Q = Q (0), the initial amount of charge on the capacitor. ( Q could be set by imposing a voltage V = Q /C across the capacitor before 1 inserting it into the circuit.) In this lab, we will use Euler’s method to numerically solve this differential equation for two different applied voltages: a constant voltage E , and then an AC voltage E ( t ) = 117 sin(120 πt ) which corresponds to the voltage out of a standard wall socket. Prelab assignment Before arriving in lab, answer the following questions. You will need your answers in lab to work the problems, and your recitation instructor may check that you have brought them. These problems are to be handed in as part of your lab report. 1. (a) Verify that the function Q 1 ( t ) = E C 1- e- t/ ( RC ) (1) is a solution to the initial value problem dQ 1 dt =- 1 RC Q 1 + 1 R E , Q 1 (0) = 0 , (2) where R , C , and E are constants. That is, by plugging t = 0 into the formula (1) show that the initial condition is satisfied, and then by differentiating the formula (1) and comparing with the right-hand side of the differential equation show that Q 1 ( t ) satisfies the differential equation. (In other words, do not try to find the solution of the initial-value problem, but rather just check that the given function solves the problem.) Then use the exact solution formula (1) with R = 12000Ω, C = 0 . 00002F,and E = 117V to complete the column labelled “Exact y ” on Table 2 on the last page of the lab, for use in Lab problem 1....
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