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To find the **gain margin** and **phase margin** of the transfer function \( G(s) = \frac{20(0.26s + 1)}{s(0.5s + 1)(0.05s + 1)} \) using a Bode plot, we proceed as follows:

1. **Rewrite the Transfer Function**:
   The transfer function is given by:
   \[
   G(s) = \frac{20(0.26s + 1)}{s(0.5s + 1)(0.05s + 1)}
   \]
   This function consists of a zero at \( s = -\frac{1}{0.26} \approx -3.846 \), a pole at the origin \( s = 0 \), and two other poles at \( s = -2 \) and \( s = -20 \).

2. **Analyze the Bode Plot**:
   - **Magnitude Plot**: The magnitude plot will give us the frequency at which the gain reaches 0 dB. The gain margin can be found at the frequency where the phase crosses \(-180^\circ\).
   - **Phase Plot**: The phase plot will help identify the frequency where the phase reaches \(-180^\circ\) and measure the phase margin at the gain crossover frequency (where the gain is 0 dB).

3. **Steps for Finding Gain and Phase Margins**:
   - **Gain Margin**: The gain margin is found by determining the frequency where the phase is \(-180^\circ\) and then observing the gain at that frequency. The gain margin (in dB) is the negative of this gain value if it’s below 0 dB.
   - **Phase Margin**: The phase margin is the amount of phase shift required to bring the phase to \(-180^\circ\) at the gain crossover frequency. The gain crossover frequency is where the magnitude is 0 dB.

4. **Using Bode Plot**:
   Using software like MATLAB, Python, or a control system toolbox, you can generate the Bode plot to obtain the exact values of the gain margin and phase margin.

Would you like a detailed step-by-step calculation using specific software, or do you have any questions about this process?

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Here are related questions to deepen understanding:

1. What is the significance of the gain margin and phase margin in control systems?
2. How does a zero in the transfer function affect the Bode plot?
3. What is the effect of adding a pole at the origin on system stability?
4. How do you interpret the stability of a system based on its gain and phase margins?
5. Can you calculate the gain and phase margins manually without using software tools?

**Tip**: When analyzing stability using a Bode plot, remember that a higher gain margin and phase margin typically indicate a more stable system.

Find the gain margin and phase margin for the system given by G(s) = 20(0.26s + 1) / s(0.5s + 1)(0.05s + 1) using a Bode plot.

Explore how to determine the gain and phase margins for a control system with transfer function G(s) = 20(0.26s + 1) / s(0.5s + 1)(0.05s + 1) using a Bode plot. Understand key concepts in control systems, including gain and phase margin calculations, and learn the significance of these parameters for system stability.

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