Spring 2021 SEEE Lecture #2 Natural and Step Responses of RLC Circuits Chapter #8 Text book: Electric Circuits James W. Nilsson & Susan A. Riedel 9th Edition. link: http://blackboard.hcmiu.edu.vn/ to download materials Mai Linh 1 Spring 2021 SEEE Objectives Be able to determine the natural response and the step response of parallel RLC circuits. Be able to determine the natural response and the step response of series RLC circuits. Outline The natural and step response of a parallel RLC circuit The natural and step response of a series RLC circuit Mai Linh 2 Spring 2021 SEEE General The natural response: The response that arise when stored energy in an inductor or capacitor is suddenly released. The step response: The response that arise when energy is being acquired by an inductor or capacitor due to sudden application of a dc voltage or current source. Second order circuits (RLC circuits): Circuits where voltages and currents are described by second-order differential equations. Mai Linh 3 Spring 2021 SEEE A second-order circuit is characterized by a second-order differential equation. • A second order circuit consists of resistors and the equivalent of two energy storage elements. • Two initial conditions (boundary conditions) are needed for solving a second-order differential equation, and those initial conditions can often be obtained via circuit analysis. Mai Linh 4 Spring 2021 Finding Initial Values Mai Linh SEEE 5 Spring 2021 SEEE Finding Initial Values (cont.) Mai Linh 6 Spring 2021 Finding Final Values Mai Linh SEEE 7 Spring 2021 Example 1 Mai Linh SEEE 8 Spring 2021 Example 1 – Sol. 12 i (0 ) = = 2 A, 4+2 i (0 + ) = i (0 − ) = 2 A − Mai Linh SEEE v(0− ) = 2i(0 − ) = 4 V v (0 + ) = v (0 − ) = 4 V 9 Spring 2021 Example 1 – Sol. Mai Linh SEEE 10 Spring 2021 Example 1 – Sol. Mai Linh SEEE 11 Spring 2021 SEEE Natural and step responses of RLC circuits Mai Linh Natural response Step response of a parallel RLC circuit. of a parallel RLC circuit. Natural response Step response of a series RLC circuit. of a series RLC circuit. 12 Spring 2021 SEEE Natural responses of a parallel RLC circuit Or the Source-Free Parallel RLC Circuit vdt Mai Linh d 2i 1 di i + + =0 2 dt RC dt LC 13 Spring 2021 SEEE Natural responses of a parallel RLC circuit Characteristic equation v = Aest , where A and s are unknown constants Characteristic equation Mai Linh 14 Spring 2021 SEEE Natural responses of a parallel RLC circuit Damping affects the way the voltage response reaches its final (steady-state) value. Mai Linh 15 Spring 2021 SEEE Natural responses of a parallel RLC circuit The over-damped response Case of: 2 0 2 s1, s2, will be real & distinct Then: v(t ) = A1e 1 + A2e st s2t To solve for A1 & A2: v(0+) = A1 + A2 Besides, dv(0+ ) and = s1 A1 + s2 A2 dt dv(0+ ) iC (0+ ) = dt C iC (0 + ) = − Mai Linh V0 − I0 R 16 Spring 2021 SEEE Natural responses of a parallel RLC circuit The over-damped response Procedure for finding the over-damped response v(t): (1) Find the roots of the characteristic equation, s1 and s2, using the values of R, L, and C. (2) Find v(0+) and dv(0+)/dt using circuit analysis. (3) Find the values of A1 and A2 by solving simultaneously equations: ( ) dv(0 ) i (0 ) = =s A +s v 0 + = A1 + A2 + + C dt C 1 1 2 A2 (4) Substitute the values for s1, s2, A1, and A2 into equation v = A1es1t + A2es2t, to determine the expression for v(t) for t ≥ 0. Mai Linh 17 Spring 2021 SEEE Natural responses of a parallel RLC circuit The under-damped response The under-damped response: v(t ) = B1e − t cos d t + B2 e − t sin d t The process for finding the under-damped response is the same as that for the overdamped response, although the response equations and the simultaneous equations used to find the constants are slightly different. Mai Linh 18 Spring 2021 SEEE Natural responses of a parallel RLC circuit The critically damped response When a circuit is critically damped, the response is on the verge of oscillating. It rarely happens in practical. Mai Linh 19 Spring 2021 SEEE under-damped = sinusoidal + exponential decay over-damped = exponential decay Mai Linh Spring 2021 The step response of a parallel RLC circuit SEEE i L + i R + iC = I v dv iL + + C =I R dt d 2iL L di L iL + + LC 2 = I R dt dt d 2iL iL 1 di L I + + = RC dt LC LC dt 2 Mai Linh If & Vf represent the final value of the response function. 21 Spring 2021 SEEE The natural response of a series RLC circuit Mai Linh d 2v R dv v + + =0 2 dt L dt LC 22 Spring 2021 SEEE The natural response of a series RLC circuit The procedure for finding the natural or step responses of a series RLC circuit are the same as those used to find the natural or step responses of a parallel RLC circuit. The neper frequency Resonant radian frequency Characteristic equation The roots The current response will be overdamped, underdamped, or critically damped according to whether Mai Linh 23 Spring 2021 SEEE The step response of a series RLC circuit Applying Kirchhoff’s voltage law Capacitor voltage step response forms in series RLC circuits 3 possible solutions for vC Mai Linh 24 Spring 2021 SEEE Example 2 Mai Linh Finding the Roots of the Characteristic Equation of a Parallel RLC Circuit 25 Spring 2021 SEEE Example 2: Solution The roots of the characteristic equation: Mai Linh 26 Spring 2021 SEEE Example 2: Solution Mai Linh 27 Spring 2021 SEEE Example 2: Solution Mai Linh 28 Spring 2021 SEEE Example 3 Finding the Overdamped Natural Response of a Parallel RLC Circuit For the circuit, v(0+) = 12V, iL(0+) = 30mA a) Find the initial current in each branch of the circuit. b) Find the initial value of dv/dt. c) Find the expression for v(t). d) Sketch v(t) in the interval 0 ≤ t ≤ 250 ms. Mai Linh 29 Spring 2021 SEEE Example 3: Solution Mai Linh 30 Spring 2021 SEEE Example 3: Solution Because the roots are real and distinct, we know that the response is overdamped. We find the co-efficients A1 & A2 ➔The overdamped voltage response: (ms) d) plot of v(t) versus t over the interval 0 t 250 ms Mai Linh 31 Spring 2021 SEEE Example 4 Solution: Mai Linh 32 Spring 2021 SEEE Example 5 Finding the Underdamped Natural Response of a Parallel RLC Circuit For the circuit, V0 = 0 V, I0 = -12.25 mA a) Calculate the roots of the characteristic equation. b) Calculate v and dv/dt at t = 0+. c) Calculate the voltage response for t ≥ 0. d) Plot v(t) for the time interval 0 Mai Linh t 11ms. 33 Spring 2021 SEEE Example 5: Solution a) Because Therefore, the response is underdamped complex frequencies b) Because v is the voltage across the terminals of a capacitor, we have v(0) = v(0+) = V0 = 0. Because v(0+) = 0, the current in the resistive branch is zero at t = 0+. Hence the current in the capacitor at t = 0+ is the negative of the inductor current: iC(0+) = -(-12.25) = 12.25 mA. Therefore the initial value of the derivative is Mai Linh 34 Spring 2021 SEEE Example 5: Solution d) The figure shows the plot of versus t for the first 11 ms after the stored energy is released. It clearly indicates the damped oscillatory nature of the underdamped response. The voltage approaches its final value, alternating between values that are greater than and less than the final value. Furthermore, these swings about the final value decrease exponentially with time. Mai Linh 35 Spring 2021 SEEE Example Mai Linh 36 Spring 2021 SEEE Example 6 Finding the Overdamped Step Response of a Parallel RLC Circuit The initial energy stored in the circuit in Fig. is zero. At t = 0 a dc current source of 24 mA is applied to the circuit. The value of the resistor is 400 . a) What is the initial value of iL? b) What is the initial value of diL/dt? c) What are the roots of the characteristic equation? d) What is the numerical expression for iL(t) when t 0? Mai Linh 37 Spring 2021 SEEE Example 6 Solution a) No energy is stored in the circuit prior to the application of the dc current source, so the initial current in the inductor is zero. The inductor prohibits an instantaneous change in inductor current; therefore iL(0) = 0 immediately after the switch has been opened. b) The initial voltage on the capacitor is zero before the switch has been opened; therefore it will be zero immediately after. Now, because v = LdiL/dt, c) From the circuit elements, we obtain roots of the characteristic equation are real and distinct Mai Linh Spring 2021 SEEE Example 6 Solution d) Because the roots of the characteristic equation are real and distinct, the inductor current response will be overdamped. Thus iL(t) takes the form of Inductor current in overdamped parallel RLC circuit step response Hence, from this solution, the two simultaneous equations that determine A’1and A’2 are The numerical solution for iL(t) is Mai Linh Spring 2021 SEEE Example 7 Finding the Underdamped Step Response of a Series RLC Circuit No energy is stored in the 100 mH inductor or the 0.4 F capacitor when the switch in the circuit shown in Fig. is closed. Find vc(t) for t ≥ 0. Mai Linh Spring 2021 SEEE Example 7: Solution Mai Linh 41 Spring 2021 SEEE Example 7: Solution Mai Linh 42 Spring 2021 SEEE Example 8 Mai Linh Spring 2021 SEEE Example 8: Sol. Mai Linh Spring 2021 SEEE Example 8: Sol. v(t) = -0.2083e-t + 5.208e-50t Mai Linh Spring 2021 SEEE Example 8: Sol. v(t) = (5 – 50t)e-10t Mai Linh Spring 2021 SEEE Example 8: Sol. v(t) = (5cos6t – 6.667sin6t)e-8t Mai Linh Spring 2021 SEEE Example 8: Sol. Mai Linh