Second Order systems II

In the underdamped case, when a unit-step input is applied, the transfer function equation is solved using the inverse Laplace method to obtain the output response.

The difference between the input and output is the error signal exhibiting a damped sinusoidal oscillation. At steady state, there is no error.

If the damping ratio equals zero, the response becomes undamped, and oscillations continue indefinitely.

In the critically damped scenario, the system's two poles are identical. For a unit-step input, the output equation and its inverse Laplace transform are determined.

In the overdamped scenario, the two poles of the system are negative, real, and unequal.

For a unit-step input, the output equation is formulated, and its inverse Laplace transformation is calculated, resulting in two decaying exponential terms.

When the damping ratio is significantly greater than unity, one exponential decay is much faster than the other and can be neglected. This results in an approximate transfer function resembling a first-order system that gives the unit step response of the system.