23.3
Second Order systems I
A servo system is an example of a second-order system. It consists of a proportional controller and load elements that align an output position with a given input position.
A second-order differential equation outlines the relationship between these elements. Applying the Laplace transform under zero initial conditions gives the transfer function, illustrating how inputs are converted to outputs.
A new interpretation of the system leads to the derivation of the closed-loop transfer function. Systems that possess two poles in this transfer function are defined as second-order systems.
The closed-loop transfer function can be rearranged and rewritten to reveal key parameters: the attenuation factor, the undamped natural frequency, and the damping ratio.
The damping ratio is defined as the ratio of actual damping to critical damping. With these parameters, the closed-loop transfer function can be rewritten in a standard form, representing the second-order system.
If the damping ratio is less than 1 but greater than 0, the system is considered underdamped.
If the damping ratio equals 1, the system is critically damped; if it's greater than 1, it is overdamped.
A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which characterizes second-order systems as those with two poles. This closed-loop transfer function can be rearranged to highlight essential parameters: undamped natural frequency and damping ratio.
The damping ratio (ζ) is a critical parameter defined as the ratio of actual damping to critical damping. It provides insight into the system's behavior and stability. With these parameters identified, the closed-loop transfer function is often rewritten in a standard form to represent the second-order system comprehensively:
Here, ωn denotes the undamped natural frequency, and ζ represents the damping ratio. The damping ratio categorizes the system's response:
Underdamped (0 < ζ < 1): The system exhibits oscillatory behavior with gradually diminishing amplitude, indicating oscillations around the equilibrium point before settling down.
Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating, often desired in servo systems for swift and smooth positioning.
Overdamped (ζ > 1): The system returns to equilibrium without oscillations but more slowly compared to a critically damped system, leading to a slower response time, which might be less desirable in applications requiring rapid adjustments.
Understanding these parameters and their influence on the system's response is crucial for designing and tuning servo systems to meet specific performance criteria. The analysis of second-order systems through their transfer functions and damping characteristics allows engineers to predict and optimize system behavior under various operating conditions.
A servo system is an example of a second-order system. It consists of a proportional controller and load elements that align an output position with a given input position.
A second-order differential equation outlines the relationship between these elements. Applying the Laplace transform under zero initial conditions gives the transfer function, illustrating how inputs are converted to outputs.
A new interpretation of the system leads to the derivation of the closed-loop transfer function. Systems that possess two poles in this transfer function are defined as second-order systems.
The closed-loop transfer function can be rearranged and rewritten to reveal key parameters: the attenuation factor, the undamped natural frequency, and the damping ratio.
The damping ratio is defined as the ratio of actual damping to critical damping. With these parameters, the closed-loop transfer function can be rewritten in a standard form, representing the second-order system.
If the damping ratio is less than 1 but greater than 0, the system is considered underdamped.
If the damping ratio equals 1, the system is critically damped; if it's greater than 1, it is overdamped.
From Chapter 23:
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23.3 : Second Order systems I
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23.1 : Transient and Steady-state Response
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23.2 : First Order Systems
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23.4 : Second Order systems II
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23.5 : Stability
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23.6 : Routh-Hurwitz Criterion I
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23.7 : Routh-Hurwitz Criterion II
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