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RL-series circuits Math 2410 Spring 2011 Consider the RL-series circuit shown in the figure below, which contains a counterclockwise current I = I(t), a resistance R, and inductance L, and a generator that supplies a voltage V (t) when the switch is closed. Kirchhoff’s voltage law satates that, “the algebraic sum of all voltage R V (t) I = I(t) L drops around an electric circuit is zero.” Applied to this RL-series circuit, the statement translates to the fact that the current I = I(t) in the circuit satisfies the first-order linear differential equation LI˙ + RI = V (t), where I˙ denotes the derivative of I with respect to t. Substituting obtain our usual form of a first-order nonhomogeneous equation: dI dt for I˙ and dividing by L, we R 1 dI + I = V (t). dt L L For convenience, we present an index of units and symbols commonly used in this theory. Quantity Unit Symbol Resistance ohm R Inductance Capacitance henry farad L C Voltage volt V Current ampere I Charge coulomb Q Time seconds sec Table 1: Table of Units Battery Resistor Inductor Capacitor Switch Table 2: Symbols of Circuit Elements Exercises 1. Use the method of integration factors to calculate what the general solution is for this differential equation. 2. In an RL-series circuit, L = 4 henries, R = 5 ohms, V = 8 volts, and I(0) = 0 amperes. Find the current at the end of 0.1 seconds. What will the current be after a very long time? 3. Assume that the voltage V (t) is given by V0 sin(ωt), where V0 and ω are given constants. Find the solution of the differential equation above subject to the initial condition I(0) = I0 . 4. Given that L = 3 henries, R = 6 ohms, V (t) = 3 sin(t), and I(0) = 10 amperes, compute the value of the current at any time t. References [1] N. Finizio and G. Ladas, Ordinary Differential Equations with Modern Applications, 3rd edition, Simon & Schuster Custom Publishing, Boston, 1999.