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.

tweede orde systemen i

Een servosysteem is een voorbeeld van een tweede-orde systeem, met een proportionele regelaar en lastenelementen die ervoor zorgen dat de uitgangspositie is uitgelijnd met de invoerpositie. De relatie tussen deze componenten wordt beschreven door een tweede-orde differentiaalvergelijking. Het toepassen van de Laplace-transformatie onder nul initiële voorwaarden levert de overdrachtsfunctie op, die laat zien hoe invoer wordt omgezet in uitvoer in het systeem.

Door het systeem opnieuw te interpreteren, kan men de gesloten-lus overdrachtsfunctie afleiden, die tweede-orde systemen kenmerkt als systemen met twee polen. Deze gesloten-lus overdrachtsfunctie kan worden herschikt om essentiële parameters te benadrukken: ongedempte eigenfrequentie en dempingsverhouding.

De dempingsverhouding (ζ) is een kritische parameter die wordt gedefinieerd als de verhouding van werkelijke demping tot kritische demping. Het biedt inzicht in het gedrag en de stabiliteit van het systeem. Met deze parameters geïdentificeerd, wordt de gesloten-lus overdrachtsfunctie vaak herschreven in een standaardvorm om het tweede-orde systeem uitgebreid weer te geven:

<img alt="Equation1" src="/files/ftp_upload/16113/16113_PT_Equation_001.svg" style="height: 44px;">

Hierbij geeft ωn de ongedempte eigenfrequentie aan en ζ vertegenwoordigt de dempingsverhouding. De dempingsverhouding categoriseert de respons van het systeem:

Ondergedempt (0 < ζ < 1): Het systeem vertoont oscillerend gedrag met geleidelijk afnemende amplitude, wat duidt op oscillaties rond het evenwichtspunt voordat het tot rust komt.

Kritisch gedempt (ζ = 1): Het systeem keert zo snel mogelijk terug naar evenwicht zonder te oscilleren, wat vaak gewenst is in servosystemen voor een snelle en soepele positionering.

Overgedempt (ζ > 1): Het systeem keert terug naar evenwicht zonder oscillaties, maar langzamer vergeleken met een kritisch gedempt systeem, wat leidt tot een langere responstijd, wat minder wenselijk kan zijn in toepassingen waarbij snelle aanpassingen vereist zijn.

Het begrijpen van deze parameters en hun invloed op de respons van het systeem is cruciaal voor het ontwerpen en afstemmen van servosystemen om te voldoen aan specifieke prestatiecriteria. De analyse van systemen van de tweede orde via hun overdrachtsfuncties en dempingskarakteristieken stelt ingenieurs in staat om systeemgedrag onder verschillende bedrijfsomstandigheden te voorspellen en optimaliseren.

Tweede orde systemen 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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Second Order systems I

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

servo-system-response-in-second-order-models

transient and steady-state response

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RC System Responses to Step Inputs

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second order systems i

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...

second order systems ii

Step Response of Second-Order Systems

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...

stability

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routh-hurwitz criterion i

Checking Stability with the Routh Table

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...

routh-hurwitz criterion ii

Routh Table Special Cases

Routh table special cases can make stability analysis harder when using the Routh-Hurwitz criterion. Two situations need extra care. One is a zero in the first column. The other is an entire row of zeros. When a single zero appears in the first column, division by zero would occur. The usual fix is to replace that zero with a very small number, epsilon (ε). The stability test then continues by assuming a sign for ε. If ε is positive, any sign change in the first column means the system is...

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.

electrical engineering

Servo System Response in Second-Order Models

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