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For problems 1-3, given a transfer function G(s) sketch the magnitude and phase characteristics in the logarithmic scale (i.e. Bode-plots) of the system using the following rules-of-thumb: i. "Normalize" the G(s) by extracting poles/zeros, substituting s-jw and writing the TF using DC-gain KO and time-constants ii. Arrange break-points (poles, zeros or wn for complex-conjugate poles) in ascending order Based on the term Ko (jw) Fk, determine: iii. a. initial slope of the magnitude-response asymptote for low frequencies as F k 20 dB/dec (e.g. flat for k=0, -20 dB/Dec for one pole at s=0 etc.) b. "anchor" point through which the magnitude-response asymptote passes for w=1 (i. e. 20 log10 Ko) C. initial value of the phase-response asymptote for low frequencies as F k 90⁰ Start going from w=0 towards ∞ iv. V. For each break-point that corresponds to a real pole/zero you encounter, adjust: magnitude-response asymptote slope by- m 20 dB/Dec for a pole and + m20 dB/Dec for a zero (m= multiplicity/order of pole/zero at the breakpoint) a. b. phase-response asymptote by - m900 for a single pole and + m 90° for a single zero (m= multiplicity/order of pole/zero at the breakpoint) vi. For each break-point wn that corresponds to a complex-conjugate set of poles/zeros, adjust: a. magnitude-response asymptote slope by -m40 dB/Dec for a set of cc poles and +m40 dB/Dec for a set of cc zeros (m= multiplicity/order of cc poles/zeros at the breakpoint) b. phase-response asymptote by - m180° for a set of cc poles and + m180⁰ for a set of cc zeros (m= multiplicity/order of cc poles/zeros at the breakpoint) vii. For <√2/2 (or 0.7) be aware of a resonant "bump" or a "dip" of approximately M₁ = i.e. My = 20 log10 dB). Draw the approximate magnitude and phase responses by fitting a curve along the asymptotes For all problems, apply and clearly indicate the bode-plot rule!. For each graph, find/indicate the approximate frequency for which the magnitude response crosses the value 0 dB. Find/indicate the approximate frequency for which the phase response crosses the value -180° (for some graphs, this will be for w->∞ ). Find/indicate Gain and Phase margins! Problem 1 G(s) = 25000 s(s+10) (s+50) (s+5)