Question

For all problems -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 i. Arange break-points (poles, zeros or on for complex-conjugate poles) in ascending order ii Based on the term Ko(ju)Fk, determine: 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.) a. b. "anchor" point through which the magnitude-response asymptote passes for w 1 (i. e. 20 log1o Ko) initial value of the phase-response asymptote for low frequencies as Fk 90° c. iv. Start going from w-0 towards For each break-point that corresponds to a real pole/zero you encounter, adjust: v. 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) phase-response asymptote by - m900 for a single pole and+ m 9 (m- multiplicity/order of pole/zero at the breakpoint) a. 00 for a single zero b.

Answer 1

Given G(s) = 1000/(s^2 + 16s + 100) rightarrow pole - zero form G(s) = 1000/[(s/10)^2 + 16s/100 + 1]100 = 10/[(s/10)^2 + 1.6(s/10) + 1] rightarrow Time constant form Here we have complex conjugate poles Natural frequency w_n = 10 rad/sec corner frequency w_n = 10 rad/sec 2G w_n = 16 damping ratio, G = 16/20 = 0.8 > 0.707 G(jw) = 10/(jw/10)^2 + 1.6(jw/10) + 1 Vertical-bar G(jw) Vertical-bar = 10/squareroot (1.6 w/10)^2 + [1 - (w/10)^2]^2 M Vertical-bar _w = 1 rad/sec = 10 = 20 log 10 = 20 dB magnitude plot: phase plot: complex conjugate poles provided a slope of - 40 dB/dec complex conjugate poles provided a phase of - 180 degree.