# Lab 5 - The Theory of RLC Circuits Eric Tao Frederick Ho...

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The Theory of RLC Circuits Eric Tao, Frederick Ho March 12th, 2013 Abstract RLC circuits are extremely important in everyday electronics, as most, if not all, circuits are made up up a combination of the simple pieces. By analyzing the behavior of these circuits in the presence of a periodic driving voltage, we hope to gain an understanding of how the circuit behavior changes with respect to time and parameters. 1 Introduction and a lot of Math We begin by considering a simple circuit made up of only an inductor, resistor, capacitor, and a driving emf. By Kirchoff’s Loop Rule, we may write an equation as such: ε - V C - V R - V I = 0 Where V denotes the voltage drop over a specific component. We know that: V C = Q/C V R = IR V I = L dI dt Where C denotes the capacitance, R denotes the resistance, L is the self-inductance, and both I and Q, the current and charge, respectively, are functions of time. Using the fact that I = dQ dt and substituting back into our loop equations, we find that: ε = L ¨ Q + R ˙ Q + 1 C Q where ¨ Q denotes the second time derivative of Q and ˙ Q denotes the first. We now note that, as L , R , and C are not functions of time, we may use the characteristic equation to solve the homogeneous case and then find our particular solutions. First setting ε = 0 and letting Q = e rt for some value of r , we substitute in to find: Lr 2 e rt + Rre rt + 1 C e rt = 0 Dividing through by

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• Winter '13
• MarceloGleiser
• Physics, Electronics, Square wave, Elementary algebra, Sine wave, RLC Circuits

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