Second Order Control System

SECOND ORDER CONTROL SYSTEM ANALYSIS
Damping : Opposition to oscillating behavior of system
It is measured by damping ratio 𝜁
In 2nd order control system, the highest power of ‘s’ in denominator of its transfer function is ‘2’.

The standard equation for Closed loop transfer function for second order control system is
𝑪(𝒔)/𝑹(𝒔) =(𝝎_𝒏^𝟐)/(𝒔^𝟐+𝟐𝜻𝝎_𝒏 𝒔+𝝎_𝒏^𝟐 )
𝜻 𝒊𝒔 𝒅𝒂𝒎𝒑𝒊𝒏𝒈 𝒓𝒂𝒕𝒊𝒐
𝝎_𝒏 𝒊𝒔 𝒖𝒏𝒅𝒂𝒎𝒑𝒆𝒅 𝒏𝒂𝒕𝒖𝒓𝒂𝒍 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝜻𝝎_𝒏 is damping factor/damping coefficient/actual damping

The denominator of its T.F. is quadratic equation and coefficients of this equations are directly related to 𝜻〖 𝒂𝒏𝒅 𝝎〗_𝒏

Characteristic Equation:
The denominator polynomial 𝒔^𝟐+𝟐𝜻𝝎_𝒏 𝒔+𝝎_𝒏^𝟐 = 0 of 𝑪(𝒔)/𝑹(𝒔) is called characteristic equation.

The roots of this characteristic equation are the poles of Transfer function 𝑪(𝒔)/𝑹(𝒔)

The roots of this characteristic equation are :

𝒔_1,𝒔_2=−𝜻𝜔_𝑛±𝒋𝜔_𝑛 √(1−𝜻^2 )
𝒔_1,𝒔_2=−𝜻𝜔_𝑛±𝒋𝜔_𝒅
Real part →Damping(𝜻𝜔_𝑛)
Imaginary part → damped frequency of oscillations 𝜔_𝒅

#SecondOrderControlSystem