We have learnt , 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 . | Previous Lecture 29 Second-Order Circuit A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. Finding Initial and Final Values We have learnt , 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 . The Source-Free Series RLC Circuit Lecture 30 Overdamped Case (α > ω0) The response is which decays and approaches zero as t increases. Critically Damped Case (α = ω0) Critically Damped Case (α = ω0) The natural response of the critically damped circuit is a sum of two terms: a negative exponential and a negative exponential multiplied by a linear term, or A typical critically damped response is shown in Fig. In fact, it is a sketch of i(t) = te−αt , which reaches a maximum value of e−1/α at t = 1/α, one time constant, and then decays all the way to zero Underdamped Case (α < ω0) Underdamped Case (α < ω0) With the presence of sine and cosine functions, it is clear that the natural response for this case is exponentially damped and oscillatory in nature. Key Points Once the inductor current i(t) is found for the RLC series circuit, other circuit quantities such as individual element voltages can easily be found. For example, the resistor voltage is vR = Ri, and the inductor voltage is vL = L di/dt. The inductor current i(t) is selected as the key variable to be determined first in order to take advantage of following Eq. because the inductor current is always continuous so that i(0+) = i(0−) Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped? Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped? Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped? Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped? Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped? Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped? Example In the following Fig., R = 40Ω, L = 4 H, and C = 1/4 F. Calculate the characteristic roots of the circuit. Is the natural response overdamped, underdamped, or critically damped?
