Understanding the Polar Plot of G(s) = 1/(1 + Ts)
The transfer function G(s) = 1/(1 + Ts) represents a first-order system. When analyzing this function in the frequency domain, we can create a polar plot that provides insights into the system's stability and response characteristics.
Key Points about the Polar Plot
- Definition of Polar Plot: A polar plot represents the magnitude and phase of a complex function as a function of frequency. For G(s), we substitute s with jω (where ω is the frequency).
- Substituting in G(s):
- G(jω) = 1 / (1 + jTω)
- Magnitude Calculation:
- The magnitude |G(jω)| is calculated as:
- |G(jω)| = 1 / √(1 + (Tω)^2)
- Phase Calculation:
- The phase angle ∠G(jω) is given by:
- ∠G(jω) = -tan^(-1)(Tω)
Shape of the Polar Plot
- Resulting Shape:
- As frequency (ω) increases, the magnitude decreases, forming a semicircular shape on the polar plot.
- The plot starts at 1 (when ω = 0) and approaches 0 as ω goes to infinity.
- Conclusion:
- The semicircle is indicative of a first-order low-pass filter behavior, confirming that the correct answer is option 'B'.
Significance of the Semicircular Shape
- Stability Insight: The semicircular nature of the plot indicates that the system is stable and demonstrates the frequency response characteristics effectively.
- Control Design: Understanding this shape aids in control system design, helping engineers predict system behavior under various input conditions.
In summary, the polar plot of G(s) = 1/(1 + Ts) is indeed a semicircle, reflecting the system's response characteristics.