tailieunhanh - Lecture Electric circuits analysis - Lecture 29: Second-order circuits

Lecture Electric circuits analysis - Lecture 29: Second-order circuits. A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. | Previous Lectures 26-28 Source free RL and RC Circuits. Unit Step Function Step Response of RC circuit Step Response of RL Circuit SECOND-ORDER CIRCUITS A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. Lecture 29 Circuit Excitation Two ways of excitation By initial conditions of the storage elements (These source free circuits will give natural responses as expected) By step inputs: Circuits are excited by independent sources. These circuits will give both the natural response and the forced response Finding Initial and Final Values We begin our lecture by learning how to obtain the initial conditions for the circuit variables and their derivatives, as this is crucial to analyze second order circuits. Perhaps the major problem students face in handling second-order circuits is finding the initial and final conditions on circuit variables. Students are usually comfortable getting the initial and final values of v and i but often have difficulty finding the initial values of their derivatives: dv/dt and di/dt . There are two key points to keep in mind in determining the initial conditions. First—as always in circuit analysis—we must carefully handle the polarity of voltage v(t) across the capacitor and the direction of the current i(t) through the inductor. Keep in mind that v and i are defined strictly according to the passive sign convention. One should carefully observe how these are defined and apply them accordingly. Second, keep in mind that the capacitor voltage is always continuous so that v(0 +) = v(0 −) (a) and the inductor current is always continuous so that i(0 +) = i(0 −) (b) where t = 0 − denotes the time just before a switching event and t = 0 + is the time just after the switching event, assuming that the switching event takes place at t = 0. Therefore, in finding initial conditions, we first focus on those variables that cannot change . | Previous Lectures 26-28 Source free RL and RC Circuits. Unit Step Function Step Response of RC circuit Step Response of RL Circuit SECOND-ORDER CIRCUITS A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. Lecture 29 Circuit Excitation Two ways of excitation By initial conditions of the storage elements (These source free circuits will give natural responses as expected) By step inputs: Circuits are excited by independent sources. These circuits will give both the natural response and the forced response Finding Initial and Final Values We begin our lecture by learning how to obtain the initial conditions for the circuit variables and their derivatives, as this is crucial to analyze second order circuits. Perhaps the major problem students face in handling second-order circuits is finding the initial and final conditions on circuit variables. Students are usually comfortable getting