Prelab Information Laboratory Preliminary Discussion Second-order RLC Circuit Analysis The second-order RLC circuit shown in figure 1 below represents all voltages and impedances as functions of the complex variable, s. Note, of course, that the impedances associated with R, RL, and Rs are constant independent of frequency, so the 's' notation is omitted. Again, one of the advantages of s-domain analysis is that we can apply all of the circuit analysis techniques learned for AC and DC circuits. ZI(s) Zc(s) Rs w RL ww + + VRS(S) VRL(S) VL(s) Vc(s) VR(S) R Vs(s) Figure 1: A second-order RLC circuit represented in the s-domain. To generate the s-domain expression for the output voltage, Vout(s) = VR(S), for the circuit shown in figure 1, we can apply voltage division in the s-domain as shown in equation 1 below. For equation 1 we define the following circuit parameters. RT=RS + RL + R where: R₁ = Total series resistance Rs Signal generator output resistance (fixed) Inductor internal resistance (fixed) R₁ R = Load resistance used as the circuit output voltage component ZL(S) = Inductive impedance as a function of the complex variable, s Zc(s) = Capacitor impedance as a function of the complex variable, s equation (1) V R R RT + Z₁ (s) + Zc(s) √s(s) Equation 1 will be used in the prelab computations to find an expression for the output voltage, VR(t), in the time domain for a specific input voltage, ys(t). Equation 1 will also be used to find an expression for the magnitude and phase angle of the transfer function defined as shown below in equation 2. equation (2) HR(s) Vout(s) VR(S) Vs(s) Vs(s) = Note also that when we collect frequency response data for the circuit it will be operating at AC steady-state conditions for each frequency tested. Note that under AC steady-state conditions, s = σ + jo = j@ = j2π£. Again, in the space provided below for prelab work, complete the following tasks for the circuit shown in Figure 1. If you require more page area to complete your computations, feel free to add space as needed between the prelab work areas below. 5. Starting with the transfer function obtained for Ha(s) in prelab part 1 above, develop expressions to generate exact Bode magnitude and phase shift plots for Ha(f). Note that approximate expressions are generally not very useful for plotting second order magnitude and phase plots for damping ratios, less than about [=0.5. Again, assume nominal values of Rs = 50[2]. R₁-110[2], R-270[2], L-50[mH], and C-5.0[NE] (5 points) 6. Use the result obtained in prelab part 5 above to generate Bode magnitude and phase shift plots via Excel or Matlab. (3 points) Generate Bode magnitude and phase shift plots using the Orcad (Papice) software to verify the frequency domain plots completed in prelab part 6 above. (1 point) 8. Compute the quality factor, Q, and bandwidth, B, for the circuit, and compare your computed values with data obtained from the plots generated in prelab parts 6 and 7 above. Again assume nominal values of Rs = 50[N], R₁ = 110[N], R = 270[N], L-50[MH], and C-5.0[DE]. (1 point)