ECE 2006 University of Minnesota Duluth Lab 9 ECE Department Page 1 April 9/11, 2007 RLC Transient Response 1. Introduction The student will analyze an RLC circuit. A unit step input will excite this circuit, producing a transient voltage response across all circuit elements. These responses will be analyzed by theory, simulation and experimental results. The primary response properties of concern are initial value and final voltage values along with voltage measurements at intermediate steps. Equations that govern selected component transient responses will be computed piecewise. To construct these equations the student will need to determine the damping coefficient, natural dampening frequency, initial capacitor voltage and final capacitor voltage. Knowing these fundamental RLC circuit properties, the student will be able to determine the roots of the circuit’s characteristic equation, basic circuit response, and corresponding transient equation. Once all these pieces are brought together to form a transient response equation, the student will compute the expected voltage at periodic time intervals. The circuit will be simulated by the student and then constructed, whereas oscilloscope measurements will be taken at intervals corresponding to those used in theoretical calculations. 2. Background A series RLC circuit may be modeled as a second order differential equation. Finding the solution to this second order equation involves finding the roots of its characteristic equation. The characteristic equation modeling a series RLC is 012=++LCLRss. This equation may be written as 02202=++ωαss with the following representations: Damping Factor = L 2R =α Undamped Natural Frequency = LC10=ω Characteristic Equation Roots = 2022,1ωαα−±−=s Knowing the above RLC circuit properties along with initial and final transient values for voltage or current then enables the transient capacitor voltage or transient inductor current to be calculated. This is done using the following equations (where infinity in this case will refer to the value when the circuit reaches steady-state before changing again): 2.1. Transient Capacitor Voltage for a Unit Step Input to a Series RLC Circuit Overdamped: α > ω0, vC(t) = vC(∞) + A1e S1t + A2e S2t V Critically Damped: α = ω0, vC(t) = vC(∞) + (A1 + A2t)e -αt V Underdamped: α < ω0, vC(t) = vC(∞) + (A1cos ωdt + A2 sin ωdt)e -αt V * *Note the difference between ωo and ωd in the equations.ECE 2006 University of Minnesota Duluth Lab 9 ECE Department Page 2 April 9/11, 2007 2.2. Transient Inductor Current for a Unit Step Input to a Series RLC Circuit Overdamped: α > ω0, iL(t) = iL (∞) + B1e S1t + B2e S2t A Critically Damped: α = ω0, iL(t) = iL (∞) + (B1 + B2t)e -αt A Underdamped: α < ω0, iL(t) = iL (∞) + (B1cos ωdt + B2 sin ωdt)e -αt A where the damped natural frequency is expressed as . In order to calculate the initial conditions, the equation relating the voltage and current through an inductor will also be needed. Since the two elements are in series, this can be rewritten as: CIdtVdLC)0()]0([+= 3. Prelab The prelab is broken down into a set of steps. For each step, you should record your answer in the appropriate table in the Data Entry section at the end of this document. Also, obtain your lab instructor’s signature before proceeding to Section 4. 3.1. Overdamped voltage transient response of capacitor in RLC circuit A. Suppose the RLC circuit in Figure 1 has component values as displayed in the figure. Assume the function generator produces a square wave with a peak-to-peak amplitude of -5 to + 5 volts, and a frequency of 50 Hz. Compute the damping factor, α, and the undamped natural frequency, ω0. Determine the initial voltage across the capacitor, vC(0+), and the final voltage across the capacitor, vC(∞). Compute the characteristic roots, 2,1s , utilizing α and ω0, and state the type of damping. B. The damping factor α should be greater than the undamped natural frequency ω0, thus the circuit is overdamped and the capacitor’s voltage transient can be expressed by the following equation with the two unknowns A1 & A2: Figure 1: RLC Circuit with 1kΩ Resistor + vR –ECE 2006 University of Minnesota Duluth Lab 9 ECE Department Page 3 April 9/11, 2007 vC(t) = vC(∞) + A1e S1t + A2e S2t Volts Evaluating at time 0, the following relationship is formed: vC(0) = _____ = vC(∞) + A1 e0 + A2 e0 Display all intermediate steps leading to a solution for A1 + A2. C. Another equation is required to solve for both unknowns, thus the derivative of the capacitor voltage transient equation is taken. dttVdC)]([ = s1A1e S1t + s2A2e S2t Again, evaluating at time 0, the following relationship is formed: dtVdC)]0([+ = C)(0iL+ = ____ = s1A1e0 + s2A2e 0 Display all intermediate steps leading to a solution for A1 & A2. D. Having solved for the two unknown variables, write the full equation for vC(t). Evaluate vC(0.5 mS), vC(1 mS) and vC(2 mS) 3.2. Underdamped voltage transient response of capacitor in RLC circuit E. Compute the damping factor, α, and the undamped natural frequency, ω0, for the circuit in Figure 2. Determine the initial voltage across the capacitor vC(0+) and the final voltage across the capacitor vC(∞). Compute the characteristic roots, 2,1s , utilizing α and ω0, and state the type of damping. Figure 2: RLC Circuit with 100Ω Resistor + vR –ECE 2006 University of Minnesota Duluth Lab 9 ECE Department Page 4 April 9/11, 2007 F. The damping factor, α, should be less than the undamped natural frequency, ω0. Thus, the circuit is underdamped and the capacitor’s transient voltage can be expressed by the following equation with the two unknowns, B1 & B2. vC(t) = vC(∞) + (B1cos ωdt + B2 sin ωdt)e -αt Volts Evaluating at time 0, the following relationship is formed: vC(0) = _____ = vC(∞) + (B1 cos(0) + B2 sin(0)) e0 Volts Display all intermediate steps leading to a solution for B1. G. Another equation is required