Bode Plots Questions and Answers

Answer: A

The frequency domain specifications of a control system are measured on the frequency response and are the resonant peak \(M_r\), the resonant frequency \(\omega_r\), the bandwidth, the cut off rate and the gain and phase margins.

The rise time is a time domain specification, measured on the step response together with the delay time, peak time, peak overshoot and settling time. So the rise time is the term that is not a frequency domain specification. The answer is option A.

Answer: A

The corner (break) frequency of each factor of the form \((1 + sT)\) in a Bode plot is \(1/T\).

\[ \begin{aligned} \omega_{c1} = \frac{1}{0.4} \\ &= 2.5\ \text{rad/s},\qquad \omega_{c2} \\ &= \frac{1}{0.1} = 10\ \text{rad/s} \end{aligned} \]

The pole at the origin, \(1/s\), has no corner frequency; it gives a line of slope \(-20\) dB per decade through the whole plot. The values 0.4 and 0.1 are the time constants, not frequencies, and 4, 1, 25 and 100 are not reciprocals of the time constants. The answer is option A.

Answer: A

The phase margin is 180 degrees plus the phase at the gain crossover frequency.

\[ PM = 180^\circ + (-150^\circ) = 30^\circ \]

The gain margin is defined from the gain at the phase crossover frequency. The item states a value of 15 dB and the key gives it as +15 dB, so the pair is \(30^\circ\) and 15 dB, option A. Strictly, a positive 15 dB gain at the phase crossover frequency would mean a gain margin of \(-15\) dB (an unstable loop); the item is loosely worded and the key takes the 15 dB as the magnitude of the margin.

The other options give a phase margin of \(-150^\circ\), \(-180^\circ\) or \(0^\circ\), which do not follow from \(-150^\circ\) at crossover. The answer is option A.