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whether the system is stable, and find the number of roots (if any) in the tight-hand s-plane. The system has no poles of G(s)G(s) in the right half-planc. (b) Deter- mine whether the system is stable if the -1 point lies at the dot on the axis.

whether the system is stable, and find the number of roots (if any) in the tight-hand s-plane. The system has no poles of G(s)G(s) in the right half-planc. (b) Deter- mine whether the system is stable if the -1 point lies at the dot on the axis.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**P9.4 Polar Plot of a Conditionally Stable System** In this section, we analyze the polar plot of a conditionally stable system, as depicted in Figure P9.4. The diagram illustrates the \( G_c(j\omega)G(j\omega) \)-plane, where \( G_c(j\omega) \) and \( G(j\omega) \) represent the transfer functions involved. The plot is used to determine stability for a specific gain \( K \). The trajectory shown on the plot approximates an oval shape that crosses the real axis at specific points, indicating potential stability issues. **Figure P9.4:** The polar plot displays a continuous path that loops around the origin, intersecting the negative real axis. It is essential for examining closed-loop stability using the Nyquist criterion. **P9.5 Engine Speed Control System** Figure P9.5 describes a block diagram of an engine speed control system. This control system consists of several components including a throttle, a torque block, and feedback loop dynamics. - **Block Diagram Details:** - Input: \( R(s) \) represents the reference input to the system. - The first block is a summation point, where the reference speed signal is compared to the feedback. - **Throttle Block:** This symbolizes the control action that adjusts engine input. - The subsequent block modifies the output with a transfer function \( \frac{1}{\tau_s s + 1} \), indicating a first-order lag. - **Torque Block:** This is represented by transfer function \( K \), showing the gain applied. - Another block with transfer function \( \frac{1}{\tau_m s + 1} \) models engine dynamics, further smoothing the response. - Output: \( Y(s) \), representing the actual speed. - A feedback loop returns a fraction of the output back to the summation point for comparison. **Figure P9.5:** It visually illustrates the feedback and control mechanism within an engine speed control system, providing insight into the relationship between various components in maintaining desired speed.
Transcribed Image Text:**P9.4 Polar Plot of a Conditionally Stable System** In this section, we analyze the polar plot of a conditionally stable system, as depicted in Figure P9.4. The diagram illustrates the \( G_c(j\omega)G(j\omega) \)-plane, where \( G_c(j\omega) \) and \( G(j\omega) \) represent the transfer functions involved. The plot is used to determine stability for a specific gain \( K \). The trajectory shown on the plot approximates an oval shape that crosses the real axis at specific points, indicating potential stability issues. **Figure P9.4:** The polar plot displays a continuous path that loops around the origin, intersecting the negative real axis. It is essential for examining closed-loop stability using the Nyquist criterion. **P9.5 Engine Speed Control System** Figure P9.5 describes a block diagram of an engine speed control system. This control system consists of several components including a throttle, a torque block, and feedback loop dynamics. - **Block Diagram Details:** - Input: \( R(s) \) represents the reference input to the system. - The first block is a summation point, where the reference speed signal is compared to the feedback. - **Throttle Block:** This symbolizes the control action that adjusts engine input. - The subsequent block modifies the output with a transfer function \( \frac{1}{\tau_s s + 1} \), indicating a first-order lag. - **Torque Block:** This is represented by transfer function \( K \), showing the gain applied. - Another block with transfer function \( \frac{1}{\tau_m s + 1} \) models engine dynamics, further smoothing the response. - Output: \( Y(s) \), representing the actual speed. - A feedback loop returns a fraction of the output back to the summation point for comparison. **Figure P9.5:** It visually illustrates the feedback and control mechanism within an engine speed control system, providing insight into the relationship between various components in maintaining desired speed.
The text discusses the stability of a system in the context of control systems and poles in the complex plane: (a) Determine whether the system is stable, and find the number of roots (if any) in the right-hand s-plane. The system has no poles of \( G_c(s)G(s) \) in the right half-plane. (b) Determine whether the system is stable if the \(-1\) point lies at the dot on the axis.
Transcribed Image Text:The text discusses the stability of a system in the context of control systems and poles in the complex plane: (a) Determine whether the system is stable, and find the number of roots (if any) in the right-hand s-plane. The system has no poles of \( G_c(s)G(s) \) in the right half-plane. (b) Determine whether the system is stable if the \(-1\) point lies at the dot on the axis.
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