Significance of the dominant poles in the feedback system of the circuit

Significance of the dominant poles

Definition of Dominant Poles:

1. Dominant Poles: These are the poles of a system's transfer function that are closest to the imaginary axis in the s-plane (Laplace domain). They have the smallest absolute values of the real parts compared to other poles of the system.

2. Location: Typically, dominant poles are located in the left half of the s-plane, indicating a stable system. Their proximity to the imaginary axis means they have a significant impact on the system's response.

Significance of Dominant Poles:

1. Transient Response:
• The dominant poles largely determine the transient response of the system, including characteristics like rise time, settling time, overshoot, and damping.
• Poles closer to the imaginary axis result in slower decay of the transient response, leading to longer settling times and potentially more overshoot.

2. Steady-State Response:
• While dominant poles primarily affect transient behavior, they also influence the steady-state accuracy, especially in systems with integral action.

3. Simplified Analysis:
• In many cases, especially in higher-order systems, the effect of non-dominant poles can be negligible for practical purposes. This allows for simplified analysis and design by focusing primarily on the dominant poles.

4. Feedback System Stability:
• In feedback systems, the stability is determined by the location of all poles, but dominant poles provide a quick insight into the system's behavior. If the dominant poles are in the left half-plane and far from the imaginary axis, the system is likely to be more stable.

5. Design and Tuning:
• In control system design, particularly in PID controller tuning, the goal is often to place the dominant poles in desired locations to achieve specific transient response characteristics.
• Adjusting feedback loop gains and adding compensators can shift the location of poles, including dominant ones, to achieve desired performance.

Example in Circuit Analysis:

Conclusion:

Proximity of poles to the imaginary axis

Understanding the s-Plane:

1. s-Plane Representation: In the s-plane, the horizontal axis represents the real part of the complex frequency , and the vertical axis represents the imaginary part . A point in this plane, represented as , corresponds to a pole of the system's transfer function.

2. Pole Location and System Behavior:
• The real part of a pole indicates the exponential growth or decay rate of the system's response. Poles with negative real parts indicate decaying responses, leading to stability.
• The imaginary part of a pole corresponds to the oscillatory component of the system's response. The closer the pole is to the imaginary axis ( approaching 0), the less damped the oscillations are, leading to a more pronounced oscillatory behavior.

Impact of Proximity to the Imaginary Axis:

1. Transient Response:
• Decay Rate: Poles close to the imaginary axis (with small ) have a slow decay rate. This results in a slower approach to the steady state, leading to longer settling times in the transient response.
• Oscillations: If these poles are complex (having a non-zero imaginary part), the system will exhibit oscillatory behavior. The closer these poles are to the imaginary axis, the less damped these oscillations are, leading to more sustained and pronounced oscillations.

2. Steady-State Response:
• While dominant poles primarily affect the transient response, they can also influence the steady-state accuracy, especially in systems with integral action or where the steady-state value depends on the system dynamics.

3. Stability Considerations:
• Poles on the imaginary axis indicate the margin of stability. If poles move into the right half of the s-plane , the system becomes unstable, exhibiting unbounded growth in its response.

4. System Sensitivity:
• Systems are generally more sensitive to changes in parameters or external disturbances when they have poles close to the imaginary axis, as these poles are indicative of lower damping and slower decay.

Conclusion: