The transfer function provided is:

$G(s) = \frac{s(s+10)(s+50)}{2000(s+0.5)}$

Steps to Analyze Stability Using the Bode Plot Approach:

1. Rewrite the Transfer Function: Simplify the given transfer function to identify its gain, poles, and zeros.

   • The numerator $(s)(s+10)(s+50)$ gives zeros at $s = 0, -10, -50$.
   • The denominator $2000(s+0.5)$ gives a pole at $s = -0.5$ and a gain factor of $\frac{1}{2000}$.

   The transfer function becomes: $G(s) = \frac{s^3 + 60s^2 + 500s}{2000(s + 0.5)}.$

2. Break Down the Magnitude Response:

   • At low frequencies ($s \to 0$), the magnitude is governed by the $(s+0.5)$ pole.
   • As frequency increases, the magnitude will be affected by the zeros and poles.
   • Each pole decreases the slope of the magnitude plot by $20\ \text{dB/decade}$, and each zero increases it by $20\ \text{dB/decade}$.

3. Phase Response:

   • Each pole contributes $-90^\circ$ to the phase shift.
   • Each zero contributes $+90^\circ$ to the phase shift.

4. Determine Stability:

   • Use the Gain Margin and Phase Margin criteria from the Bode plot to assess stability.
   • A system is stable if the gain margin and phase margin are both positive.

5. Draw the Bode Plot:

   • Use the transfer function to calculate the magnitude and phase for a range of frequencies (typically logarithmic scale).
   • Plot the results to analyze the stability.

Would you like a detailed plot or step-by-step analysis of the magnitude and phase responses?

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Related Questions:

1. How can poles and zeros affect the system's stability in the frequency domain?
2. What is the significance of the gain margin and phase margin in determining stability?
3. How can the Nyquist criterion complement the Bode plot approach in stability analysis?
4. What changes in the transfer function $G(s)$ would make the system unstable?
5. How does scaling the gain ($2000$) affect the Bode plot and system stability?

Tip:

To simplify Bode plots, break the system into first-order and second-order components and analyze each separately.