What is a Second Order System?
A second-order system is a powerful framework portrayed by a second-degree transfer function in the Laplace domain. The general type of a second-order transfer function is addressed as:
G(s) = K / (s-a) (s-b)
where ,
K is the system gain,
s is the complex frequency variable, and
a and b are the system poles.
Derivation of Second Order System
To derive the transfer function of a 2nd-order system, remember an ordinary dynamic machine represented via a mass-spring-damper device. This system includes a mass m related to a spring with spring steady k and a damper with damping coefficient c. The input to the device is a force F(t), and the output is the displacement x(t) of the mass.
Applying Newton’s second law of motion to the mass yields the following second-order ordinary differential equation (ODE):
md2x / dt2 + cdx/dt + kx = F(t)
here we are Taking the Laplace transform of both sides of the above equation, so we obtain:
ms2 X(s) + csX(s) + KX(s) = F(s)
here the Laplace transforms of x(t) and F(t) are X(s) and F(s).
after this we will getting:
(ms2 + cs + k) X(s) = F(s)
X(s)/F(s) = 1 / ms2 + cs + k
This expression represents the transfer function G(s) of the second-order system, where:
K = 1/m
x = -c/2m + √c2 – 4mk /2m
y = -c/2m – √c2 -4mk / 2m
The poles x and y of the transfer function determine the nature of the system’s response: whether it is overdamped, underdamped, or critically damped.
Response of Second Order System
Control systems play a critical position in regulating and keeping the conduct of dynamic structures, making sure of balance and desired overall performance. One common form of machine encountered in the control idea is the second one-order system. The reaction of such structures is essential to understand for engineers and researchers operating in various fields. Now let’s move on to the concepts of pole and zero and the transient response to the second order system.
In contrast to the simplicity of first-order systems, second-order systems have many answers that need to be analyzed and explained. Changing first-order parameters only changes the response rate, while changing second-order parameters can change the response. For example, the second order may show similar behavior to the first order, or it may show temporary responses, either negative or weak, depending on the value of the product. In this article, we delve into the traits, analysis, and importance of the response of the second-order system on top of things theory.
Table of Content
- Second Order System
- Characteristics
- Step Response
- Transient Response Specification
- Types
- Mathematical Formula
- Importance
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