Final answer:
To sketch the Bode plot for L(s) when k = 5, we need to analyze the transfer function and make qualitative observations. The magnitude plot starts at 0 dB, decreases by 20 dB/decade until the break frequency, then decreases more rapidly at -40 dB/decade. The phase plot starts at 0°, decreases to -90°, makes a sharp transition to -180° at the break frequency, and continues to decrease beyond that point.
Step-by-step explanation:
To sketch the Bode plot for L(s) when k = 5, we need to calculate the transfer function of L(s). The transfer function is given by:
L(s) = Gc(s) G(s) H(s) = (k / s(s^2 + s + 9)) * 1 = k / (s(s^2 + s + 9))
Since k = 5, the transfer function becomes:
L(s) = 5 / (s(s^2 + s + 9))
To sketch the Bode plot, we need to analyze the magnitude and phase of L(s) for different frequencies. We can do this by expressing L(s) in the form of an ordinary differential equation and using techniques such as partial fraction decomposition and Laplace transforms. However, since the question only asks for a sketch, we can make some qualitative observations without performing detailed calculations.
From the transfer function, we can see that L(s) has a pole at s = 0 and two complex poles at s = -0.5 ± 3.354i. The pole at s = 0 contributes a slope of -20 dB/decade to the magnitude plot, while the complex poles contribute slopes of -40 dB/decade and a phase lag of -90°. Therefore, the magnitude plot starts at 0 dB and decreases by 20 dB/decade until it reaches the break frequency ω = 3.354 rad/s. At this point, the magnitude plot starts to decrease more rapidly, with a slope of -40 dB/decade, until it reaches -∞ dB. The phase plot starts at 0° and gradually decreases to -90° as the frequency increases. At the break frequency ω = 3.354 rad/s, the phase plot makes a sharp transition to -180° and continues to decrease beyond that point.