(a) ta > 4.6s, tr> 2.2s, %OS < 16% (b) t < 1.1s (c) t < 1.15s, tr< 0.55s d) ta < 0.66s, tr< 0.314s, % OS < 5%.
(a) ta > 4.6s, tr> 2.2s, %OS < 16% (b) t < 1.1s (c) t < 1.15s, tr< 0.55s d) ta < 0.66s, tr< 0.314s, % OS < 5%.
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Each plot's solution has been given, please show the steps required to obtain each solution.
Transcribed Image Text:(a) \( t_s > 4.6s, t_r > 2.2s, \%OS < 16\% \)
(b) \( t_r < 1.1s \)
(c) \( t_s < 1.15s, t_r < 0.55s \)
(d) \( t_s < 0.66s, t_r < 0.314s, \%OS < 5\% \)
**Explanation:**
- \( t_s \): Settling time
- \( t_r \): Rise time
- \%OS: Percent overshoot
Each option specifies conditions for system responses, such as settling time, rise time, and percent overshoot. These metrics are critical in analyzing system dynamics and stability in control systems.
![**Title: Second Order System Analysis**
**Transfer Function of Second Order System:**
\[
T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2}
\]
**Objective:**
Analyze each plot to determine guarantees on system performance characteristics, such as step response rise time, settling time, and percent overshoot. Specifically, we seek inequalities like \( t_s < 3 \, s \).
**Plots Explanation:**
**Plot (a):**
- **Description:** The shaded region is a sector in the left half of the s-plane, bounded by an angle of 60° from the negative real axis. It extends from the origin outward.
- **Interpretation:** If poles are within this sector, they reflect a specific range of damping ratio and natural frequency, influencing the transient response of the system.
**Plot (b):**
- **Description:** The shaded area is a semicircular region centered on the real axis with a radius extending up to \( j2 \) from the origin.
- **Interpretation:** Poles within this semicircle indicate constraints on the natural frequency and damping ratio, impacting the speed of response and stability.
**Plot (c):**
- **Description:** The shaded region is a vertical band extending from the real part of \(-4\) and beyond to the left.
- **Interpretation:** This plot suggests if the poles are within this band, certain assurances on the maximum settling time can be made due to the pole locations far from the imaginary axis.
**Plot (d):**
- **Description:** The shaded region is a triangular area bounded by two lines: the vertical line through \(-1\) on the real axis and a line at 45° to the real axis.
- **Interpretation:** Poles in this triangle indicate a balance between rise time and overshoot, providing certain behavioral guarantees for the system's transient response.
**Conclusion:**
Analyzing pole locations within these regions helps predict system behavior, facilitating control design to meet specified performance metrics. Understanding these regions is crucial for determining the permissible ranges of system parameters, directly affecting stability and transient performance.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2de2484e-8c30-4a24-aa43-7ad85b110ab5%2Ff13cb7a3-4c2c-491b-9ee3-e6643f4e8ce2%2Fh8k1rnb_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Second Order System Analysis**
**Transfer Function of Second Order System:**
\[
T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2}
\]
**Objective:**
Analyze each plot to determine guarantees on system performance characteristics, such as step response rise time, settling time, and percent overshoot. Specifically, we seek inequalities like \( t_s < 3 \, s \).
**Plots Explanation:**
**Plot (a):**
- **Description:** The shaded region is a sector in the left half of the s-plane, bounded by an angle of 60° from the negative real axis. It extends from the origin outward.
- **Interpretation:** If poles are within this sector, they reflect a specific range of damping ratio and natural frequency, influencing the transient response of the system.
**Plot (b):**
- **Description:** The shaded area is a semicircular region centered on the real axis with a radius extending up to \( j2 \) from the origin.
- **Interpretation:** Poles within this semicircle indicate constraints on the natural frequency and damping ratio, impacting the speed of response and stability.
**Plot (c):**
- **Description:** The shaded region is a vertical band extending from the real part of \(-4\) and beyond to the left.
- **Interpretation:** This plot suggests if the poles are within this band, certain assurances on the maximum settling time can be made due to the pole locations far from the imaginary axis.
**Plot (d):**
- **Description:** The shaded region is a triangular area bounded by two lines: the vertical line through \(-1\) on the real axis and a line at 45° to the real axis.
- **Interpretation:** Poles in this triangle indicate a balance between rise time and overshoot, providing certain behavioral guarantees for the system's transient response.
**Conclusion:**
Analyzing pole locations within these regions helps predict system behavior, facilitating control design to meet specified performance metrics. Understanding these regions is crucial for determining the permissible ranges of system parameters, directly affecting stability and transient performance.
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