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RLC Circuit Response

Analyzes the transient and steady-state behavior of circuits containing resistors (R), inductors (L), and capacitors (C). The governing equation is a second-order differential equation.
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The statement of the theorem

Let i(t)i(t) be the current flowing through a series RLC circuit, and let vin(t)v_{in}(t) be the applied voltage source. The circuit response is governed by the second-order linear non-homogeneous ordinary differential equation (ODE) derived from Kirchhoff's Voltage Law (KVL): \n\nddt(Ldidt)+Rdidt+1Ci=ddtvin(t)or, more commonly, using the charge q(t)=∫i(t)dt:\frac{d}{dt}\left(L \frac{di}{dt}\right) + R \frac{di}{dt} + \frac{1}{C} i = \frac{d}{dt} v_{in}(t) \quad \text{or, more commonly, using the charge } q(t) = \int i(t) dt: \n\nLd2qdt2+Rdqdt+1Cq=vin(t)L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = v_{in}(t) \n\nwhere LL, RR, and CC are the inductance, resistance, and capacitance, respectively. The solution q(t)q(t) is decomposed into the homogeneous solution qh(t)q_h(t) (transient response) and the particular solution qp(t)q_p(t) (steady-state response): \n\nq(t)=qh(t)+qp(t)q(t) = q_h(t) + q_p(t) \n\nThe characteristic equation for the transient response is λ2+RLλ+1LC=0\lambda^2 + \frac{R}{L}\lambda + \frac{1}{LC} = 0, determining the damping regime (overdamped, critically damped, or underdamped).