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Time Response of Second Order System | Control Systems - Electrical Engineering (EE) PDF Download

Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

In the above transfer function, the power of 's' is two in the denominator. That is why the above transfer function is of a second order, and the system is said to be the second order system.

Time response of second order system with unit step

From equation 1

Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
For unit step the input is

Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Now taking the inverse Laplace of above equation
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
This equation can also be written as
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
The error signal is given by e(t) = r(t) - c(t), and hence
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

Transient response specification of second order system

The performance of the control system are expressed in terms of transient response to a unit step input because it is easy to generate initial condition basically are zero.
Following are the common transient response characteristics:

  • Delay Time.
  • Rise Time.
  • Peak Time.
  • Maximum Peak.
  • Settling Time.
  • Steady State error.
  • Delay Time

The time required for the response to reach 50% of the final value in the first time is called the delay time.

Rise Time
The time required for response to rising from 10% to 90% of final value, for an overdamped system and 0 to 100% for an underdamped system is called the rise time of the system.

Peak Time
The time required for the response to reach the 1st peak of the time response or 1st peak overshoot is called the Peak time.

Maximum overshoot
The difference between the peak of 1st time and steady output is called the maximum overshoot. It is defined by
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

Settling Time (ts)
The time that is required for the response to reach and stay within the specified range (2% to 5%) of its final value is called the settling time.
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

Steady State Error (ess)
The difference between actual output and desired output as time't' tends to infinity is called the steady state error of the system.
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

Example
When a second-order system is subjected to a unit step input, the values of ξ = 0.5 and ωn = 6 rad/sec. Determine the rise time, peak time, settling time and peak overshoot.

Given-
ξ = 0.5 ω n = 6 rad/sec  
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Peak Time:
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)
Settling Time:
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

Maximum overshoot:
Time Response of Second Order System | Control Systems - Electrical Engineering (EE)

The document Time Response of Second Order System | Control Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Control Systems.
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FAQs on Time Response of Second Order System - Control Systems - Electrical Engineering (EE)

1. What is a second order system?
Ans. A second order system is a type of dynamic system that can be described by a second order differential equation. It consists of two poles and represents a system with two energy storage elements, such as mass-spring systems or RLC circuits.
2. How is the time response of a second order system defined?
Ans. The time response of a second order system refers to how the system behaves or reacts to an input over time. It can be characterized by parameters such as rise time, settling time, peak time, and overshoot.
3. What are the characteristics of the time response of a second order system?
Ans. The time response of a second order system depends on its damping ratio. For a critically damped system, it has no overshoot and reaches its final value as quickly as possible. For an overdamped system, it has no overshoot but takes longer to reach its final value. For an underdamped system, it oscillates before settling to its final value and may have overshoot.
4. How is the damping ratio of a second order system related to its time response?
Ans. The damping ratio of a second order system determines the type of response it exhibits. A higher damping ratio leads to a slower and smoother response with less overshoot, while a lower damping ratio leads to a faster response with more oscillations and potentially larger overshoot.
5. What are the applications of second order systems in real life?
Ans. Second order systems are commonly found in various engineering disciplines. They are used in mechanical systems like car suspensions, electrical systems like amplifiers and filters, and control systems for robotics and industrial processes. Understanding their time response is crucial for designing and analyzing these systems.
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