Bode Plots Questions and Answers

Answer: D

The key takes the assertion as false and the reason as true, the fourth option. The stability of a system does not depend on how near the poles are to the origin, and the condition is that the poles lie in the left half of the s-plane, so the assertion is false.

The key accepts the reason on zeros as the true statement. The options which accept the assertion, the first, second and third, are wrong, since the assertion about poles being near the origin does not give stability.

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.

Answer: A

The slope is 20 dB/decade.

The magnitude of \(j\omega\) is \(\omega\), so on the log magnitude plot it contributes \(20\log_{10}\omega\) dB. Each tenfold increase in \(\omega\) adds 20 dB: a straight line of slope +20 dB/decade passing through 0 dB at \(\omega = 1\).

6 dB/decade confuses the decade with the octave: +20 dB/decade is the same as about +6 dB/octave. The negative slopes, −20 dB/decade and −6 dB/octave, belong to the pole \(1/j\omega\), not to the zero \(j\omega\).

Answer: C

The gain margin is 10.

Gain margin is how much the loop gain can be multiplied at the phase crossover frequency before the system reaches instability. If the gain there is -20 dB, the gain margin is +20 dB. Converting decibels to a magnitude ratio:

\[ 20 \log_{10} \text{GM} = 20 \Rightarrow \text{GM} = 10 \]

Equivalently, the loop gain at phase crossover is 0.1, and the gain margin is its reciprocal, 10. 20 is the value in decibels, not the ratio; 1 would mean the system is on the edge of instability, and 5 has no basis.

Answer: A

The key takes both statements as true with (R) explaining (A), option A.

(A) is a fair statement of practice: most linear control systems are stable, which puts their poles in the left half of the s-plane, and the zeros of most plants lie there too. (R) is the standard definition: a transfer function with no poles or zeros in the right half s-plane is called minimum phase. So (R) names the class that (A) describes, and that is the sense in which the key treats it as the explanation.

Whether naming the class counts as explaining why many systems belong to it is a matter of reading, and I could not settle it from a published source; I found nothing that shows the key wrong, so it is kept. Options C and D are wrong because both statements are true: (R) is the textbook definition of minimum phase.

Answer: A

The correct matching is a-iii, b-iv, c-ii, d-i.

• The Bode plot is a frequency response method, plotting magnitude and phase against frequency (iii).
• The Nyquist criterion determines the stability of a closed-loop control system from its open-loop frequency response (iv).
• The Z-transform is the tool for sampled-data (discrete-time) systems (ii).
• The inverse Laplace transform gives the time response solution from the s-domain expression (i).

The other codes give the Bode plot or the Nyquist criterion each other's role, or the Z-transform the time response.

Answer: B

The Bode plot is used to visualize the frequency response of a system.

It is a pair of graphs drawn against log frequency: gain in decibels, \(20\log_{10}|G(j\omega)|\), and phase angle. Because each pole and zero adds a straight-line asymptote on these axes, the plots can be sketched quickly, and they give the bandwidth, corner frequencies, and the gain and phase margins used to judge closed-loop stability.

The step response, transient behaviour and time-domain behaviour are all time-domain descriptions, obtained from the differential equation or the inverse Laplace transform, not from a frequency-response plot. That rules out the other three options.

Answer: C

Such a system is oscillatory.

Gain margin is how much the loop gain can increase before instability, and phase margin is how much extra phase lag can be tolerated. A gain margin close to unity (0 dB) or a phase margin close to zero means the open-loop frequency response passes very near the critical point -1 in the Nyquist plot. The closed-loop system then has poles close to the imaginary axis, so its response is lightly damped and oscillates for a long time; at exactly unity gain margin or zero phase margin it oscillates continuously.

A highly or relatively stable system has comfortable margins, typically a gain margin of several dB and a phase margin of 30-60°. An unstable system has a negative margin.