Consider a unity feedback system with its forward transfer function by G(s) = K/[s(s2+4s+5)] K > 0. (a) Find the asymptotes and sketch them in the s-plane, (b) Determine the angle of departure from the complex poles, (c) Find the breakaway points (if any), (d) Determine the jw- axis crossing points and the corresponding gain K, and (e) Sketch the root locus.

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**Problem Description: Unity Feedback System Analysis**

Consider a unity feedback system with its forward transfer function given by:

\[ G(s) = \frac{K}{s(s^2 + 4s + 5)} \]

where \( K \geq 0 \).

1. **Asymptotes and Sketch in the s-plane:**
   - Find the asymptotes and represent them graphically in the s-plane.

2. **Angle of Departure:**
   - Calculate the angle of departure from the complex poles.

3. **Breakaway Points:**
   - Identify the breakaway points, if any.

4. **jω-axis Crossing Points and Corresponding Gain K:**
   - Determine the points where the root locus crosses the jω-axis and find the corresponding gain \( K \).

5. **Root Locus Sketch:**
   - Draw a sketch of the root locus.

The topics covered include understanding the transfer function of a control system, analyzing pole and zero locations, and constructing the root locus to study system stability and performance. This problem emphasizes step-by-step derivation and graphical representation techniques used in control system design and analysis.
Transcribed Image Text:**Problem Description: Unity Feedback System Analysis** Consider a unity feedback system with its forward transfer function given by: \[ G(s) = \frac{K}{s(s^2 + 4s + 5)} \] where \( K \geq 0 \). 1. **Asymptotes and Sketch in the s-plane:** - Find the asymptotes and represent them graphically in the s-plane. 2. **Angle of Departure:** - Calculate the angle of departure from the complex poles. 3. **Breakaway Points:** - Identify the breakaway points, if any. 4. **jω-axis Crossing Points and Corresponding Gain K:** - Determine the points where the root locus crosses the jω-axis and find the corresponding gain \( K \). 5. **Root Locus Sketch:** - Draw a sketch of the root locus. The topics covered include understanding the transfer function of a control system, analyzing pole and zero locations, and constructing the root locus to study system stability and performance. This problem emphasizes step-by-step derivation and graphical representation techniques used in control system design and analysis.
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