Name: servo system response in second-order models
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 20 s
Channel: JoVE

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&#39;s greater than 1, it is overdamped.

second order systems i

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 (&#950;) 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:
<img alt="Transfer function equation, second-order system formula, control systems analysis." src="/files/ftp_upload/16113/16113_PT_Equation_001.svg" style="height: 44px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Here, &#969;n denotes the undamped natural frequency, and &#950; represents the damping ratio. The damping ratio categorizes the system's response:
Underdamped (0 &#60; &#950; &#60; 1): The system exhibits oscillatory behavior with gradually diminishing amplitude, indicating oscillations around the equilibrium point before settling down.
Critically damped (&#950; = 1): The system returns to equilibrium as quickly as possible without oscillating, often desired in servo systems for swift and smooth positioning.
Overdamped (&#950; &#62; 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.

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 ...

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servo system response in second-order models

A servo system is a second-order model that links an input position to an output position. It uses a proportional controller and load elements to keep the output aligned with the input. This behavior can be written as a second-order differential equation.

When the equation is transformed with the Laplace transform under zero initial conditions, it becomes a transfer function. The transfer function shows how the input is converted into the output. It can also be rewritten as a closed-loop transfer function, which is a common way to describe a second-order system.

Second-order systems have two poles. Their closed-loop transfer function is often rearranged to show two key parameters: the undamped natural frequency and the damping ratio. The damping ratio, written as zeta, is the ratio of actual damping to critical damping. It helps describe system behavior and stability.

The standard form of the closed-loop transfer function uses omega n for the undamped natural frequency and zeta for the damping ratio. These values help explain how the system responds over time. They are also useful when designing and tuning servo systems for specific performance goals.

The damping ratio divides the response into three cases. An underdamped system has 0 1 and does not oscillate, but it returns to equilibrium more slowly than a critically damped system.

Servo System Response in Second-Order Models

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A servo system is a second-order model that links an input position to an output position. It uses a proportional controller and load elements to keep the output aligned with the input. This behavior can be written as a second-order differential equation. When the equation is transformed with the Laplace transform under zero initial conditions, it becomes a transfer function. The transfer function shows how the input is converted into the output. It can also be rewritten as a closed-loop...

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Second-order systems react to a unit-step input in different ways depending on the damping ratio. When the damping ratio, zeta, is between 0 and 1, the system is underdamped. In this case, inverse Laplace methods give an output that oscillates with a decreasing amplitude. The error signal, which is the difference between the input and output, also shows damped oscillations before it goes to zero at steady state. When zeta equals 0, the system is undamped. The response keeps oscillating forever...

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Checking stability with the Routh table is a useful part of analyzing a closed-loop transfer function. It helps determine whether a system will remain stable when load conditions change or when a fault occurs. In a power grid, this kind of check is important because instability can lead to blackouts. The Routh-Hurwitz criterion uses a table built from the denominator polynomial of the transfer function. The rows of the Routh table are labeled by powers of the complex variable s, starting with...

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key terms and concepts

Learn how servo systems act as second-order models, how transfer functions are formed, and how damping ratio shapes underdamped, critical, and overdamped response.

Servo System Response in Second-Order Models | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

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Servo System Response in Second-Order Models

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