Second-Order Circuits: Computing Inductor Current i(t) in RLC Circuits, Study notes of Signals and Systems

Download Second-Order Circuits: Computing Inductor Current i(t) in RLC Circuits and more Study notes Signals and Systems in PDF only on Docsity! EECS 216 2nd-ORDER CIRCUITS: SINUSOIDAL SOURCES NATURAL RESPONSE (NO SOURCE PRESENT) Given: Series RLC circuit (resistor+inductor+capacitor connected together). Initial: Capacitor is charged up to v(0) and inductor is “juiced” up to i(0). t=0: Close or throw the switch at t = 0. Recall i(t) and v(t) don’t jump. Goal: Compute current through inductor (and everything else) i(t) for t > 0. Devices: i = C dvdt → v(t) = 1C ∫ t −∞ i(t ′)dt′ and v = Ldidt → i(t) = 1L ∫ t −∞ v(t ′)dt′. KVL: 1C ∫ t −∞ i(t ′)dt′ + Ldidt + iR = 0. Take 1 L d dt → d 2i dt2 + R L di dt + 1 LC i = 0. Note: Any circuit variable satisfies this differential equation. So use the fol- lowing procedure to compute inductor current iL(t) or capacitor volt- age vC(t) (usually easiest), then compute other quantities from those. Soln: The trial solution i(t) = Iest solves the above differential equation if [s2 + RL s + 1 LC ]Ie st = 0 → s2 + RL s + 1LC = 0 (characteristic equation) So: i(t) = C1es1t+C2es2t for some constants C1, C2, s1, s2 (Math 216 idea) where: C1 and C2 determined by initial conditions v(0) and i(0) and: s1 and s2 are the two roots of the characteristic equation. Three different cases: Overdamped, underdamped, critically damped: 1. ( R 2L )2 > 1 LC → s1,2 = − R2L ± √ ( R 2L )2 − 1 LC < 0 (so it always decays) 2. ( R2L ) 2 < 1LC → s1,2 = − R2L ± j √ 1 LC − ( R2L )2 = −α ± jωd where: α = R2L and ωd = √ 1 LC − ( R2L )2 = the damped natural frequency. Note: 1α = 2 L R acts like a time constant, affecting amplitude of sinusoids. 3. ( R2L ) 2 = 1LC → i(t) = C1e−αt + C2te−αt (called critical damping). COMPLETE RESPONSE (NOW INCLUDE SOURCE) Formula: i(t) = iSS(t)︸ ︷︷ ︸ steady−state + C1es1t + C2es2t︸ ︷︷ ︸ transient decays to 0 for t > 0. where: C1 and C2 are determined by the initial conditions v(0) and i(0). Note: Follows from linearity of differential equation; C1 and C2 are deter- mined by the initial conditions, used to determine i(0+) and di dt (0+).