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ECE 2006 LABORATORY 8RLC TRANSIENT RESPONSEOBJECTIVESThe learning objectives for this laboratory are to give the student the ability to:- use the function generator to generate a step input with an appropriate repetition rate.- use the oscilloscope to measure RLC overdamped and underdamped transient response values.- use the PSPICE transient response tool.REFERENCES- Alexander/Sadiku, “Fundamentals of Electric Circuits – 2nd Edition”, 2003, McGraw-HillBACKGROUND- See above reference, Appendix D.4: pp.A-38 thru A-46, for PSPICE analysis of RLC transient response.- The analysis of series RLC circuits can be summarized as follows:Finding the solution to a second order linear differential equation with a unit-step forcing function involves finding the roots of its characteristic equation. The characteristic equation is quadratic and thus has two roots. The roots may be 1) real and equal, 2) real but unequal or 3) complex conjugates.A series RLC circuit can be modeled as a second order differential equation. When its roots are real and equal, the circuit response to a step input is called “Critically Damped”. When its roots are real but unequal the circuit response is “Overdamped”. When roots are a complex conjugate pair, the circuit repsonse is labeled “Underdamped”.To find the solutions for voltage and current in an RLC circuit, such as in Figure A, it is Figure A: Series RLC Circuit with Step Inputnecessary to first determine its damping by finding the roots of the characteristic equation:Series RLC Characteristic Equation: S2 + S(R/L) + 1/LC = 0Putting the above equation into standard form: S2 + 2αS + ω02 = 0, it follows that:α = R/2L and ω0 = 1 / √LCwhere: α is the Damping Coefficient andω0 is the Natural or Resonant FrequencyThe roots of the Characteristic Equation, by using the Quadratic Formula, are:S1,2 = - α ± √(α2 - ω02)The damping is determined by the ratio of α/ω0. If the ratio is greater than one, i.e. α > ω0, the circuit is Over Damped. If α = ω0, the circuit is Critically Damped. Otherwise, the circuit is UnderDamped. For each type of damping condition, the voltage and current solutions take a different form:CAPACITOR VOLTAGE FOR A STEP INPUT TO A SERIES RLC CIRCUIT:Overdamped: α > ω0, Vc(t) = Vc(∞) + A1e S1t + A2e S2t VoltsCritically Damped: α = ω0 , Vc(t) = Vc(∞) + (A1 + A2t)e -αt VoltsUnderdamped: α < ω0 , Vc(t) = Vc(∞) + (A1cos ωdt + A2 sin ωdt)e -αt Volts ωd = √(α2 - ω02)INDUCTOR CURRENT FOR A STEP INPUT TO A SERIES RLC CIRCUIT:Overdamped: α > ω0, IL(t) = IL (∞) + B1e S1t + B2e S2t AmpsCritically Damped: α = ω0 , IL (t) = IL (∞) + (B1 + B2t)e -αt AmpsUnderdamped: α < ω0 , IL (t) = IL (∞) + (B1cos ωdt + B2 sin ωdt)e -αt Amps ωd = √(α2 - ω02)Once the damping condition is known and the form of the solution is determined, it necessary only to determine the values of the coefficients, A1, A2, B1, and B2 in order to form a complete solution.It requires two independent equations to solve for two unknowns. Since it is possible to determine initial and final values for Capacitor Voltage and Inductor Current, evaluating the capacitor voltage equation and its first derivative at time, t = 0, will form an independent pair of equations:Find Vc(0), Vc(∞) by inspecting the circuit.Find d[Vc(0)]/dt by using the equation, Ic(0+) = C dVc(0+)/dtSolve for A1, A2:For example, in an overdamped circuit with Zero initial conditions:Vc(0) = 0 = Vc(∞) + A1e 0 + A2e 0d[Vc(0)]/dt = 0 = S1A1e 0 + S2A2e 0Solve the two equations simultaneously to find the A constants.EQUIPMENT OscilloscopeFunction generatorResistor, 100 -Resistor, 1.0 k-Capacitor, 1.0 -FInductor, 220 mHPROCEDURE1. Overdamped RLC circuit capacitor voltage transient response to a step input.1.1 With the RLC circuit disconnected, adjust the function generator to produce a repetitive pulse that is -5 volts for about 10 msec, then +5 volts for about 10 msec. (i.e. 10 Volts peak-to-peak, 0 Volts of DC offset, 20 msec Period or 50 Hz)1.2 For the circuit in Figure 1, calculate the output response, VC(t), t > 0, to an input step, from -5 to +5 Volts.FIGURE 1: Series RLC with Step Input, Measuring VcFirst determine α and ω0. Calculate the roots of the characteristic equation, S1,2. Determine Vc(0), Vc(∞) , and d[Vc(0)]/dt. Calculate A1 and A2. Fill in the calculated values in the Data Table for Figure 1 below:DATA TABLE 1: OVERDAMPED RLCQuantity Calculated Value Measured ValueαN/Aω0N/AOver,Under orCritical Damping?S1,2N/AVc(0)Vc(∞)d[Vc(0)]/dt = IL(0+)/C = 0.000 Amps/CoulombA1 and A2N/AEquation for Vc(t): N/AVc(0.5ms):Vc(1.0ms):Vc(2.0ms):1.3 Connect the circuit in Figure 1. Measure the final value, VC(t=-), and the initial value, VC(t=0+), from the oscilloscope and record in the Data section. Also measure the voltagesVC(t=0.5 msec), VC(t=1.0 msec), and VC(t=2.0 msec)from the oscilloscope and record in the Data section.2. Underdamped RLC circuit capacitor voltage transient response to a step input.2.1 Keep the function generator settings used in Part 1.2.2 For the circuit in Figure 2, calculate the output response, VC(t), t > 0, to an input step, from -5 to +5 Volts. Note that the only change to the circuit is replacing the 1 k-Ohm resistor with a 100 Ohm resistor.FIGURE 2: Underdamped RLC Circuit with Step InputFirst determine α and ω0. Calculate the roots of the characteristic equation, S1,2. Calculate ωd.Determine Vc(0), Vc(∞) , and d[Vc(0)]/dt. Calculate A1 and A2. Fill in thecalculated values in the Data Table for Figure 2 below:DATA TABLE 2: UNDERDAMPED RLCQuantity Calculated Value Measured ValueαN/Aω0N/AOver,Under orCritical Damping?S1,2 and ωdContinued on NEXT PAGEωd = 2π/T = Vc(0)Vc(∞)d[Vc(0)]/dt = IL(0+)/C = 0.000 Amps/CoulombA1 and A2N/AEquation for Vc(t):Vc(0.5ms):Vc(1.0ms):Vc(2.0ms):2.3 Connect the circuit in Figure 2. Measure the final value, VC(-), the initial value, VC(0+), and T/2 (one half the period of oscillation of the output waveform) from the oscilloscope and record in the Data section. Also measure the voltages V2(t=0.5msec), V2(t=1.0msec), and V2 (t=2.0msec) from the oscilloscope and record in the Data section.3. Underdamped RLC circuit resistor voltage transient response to a step input.3.1 Keep the function generator settings used in Part 2.3.2 For the circuit in Figure 3, calculate the output response, VR(t), t > 0, to an input step, from -5 to


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