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)
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Chapter1: Introduction
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb6e49c2-d01d-4438-a99a-92f670cec770%2F496fc290-8885-45dd-86a3-f6ed36ca4161%2Fx1ti5oh_processed.png&w=3840&q=75)
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb6e49c2-d01d-4438-a99a-92f670cec770%2F496fc290-8885-45dd-86a3-f6ed36ca4161%2Fbyiulc_processed.png&w=3840&q=75)
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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