# Time response of second order system

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

For unit step the input is

Now taking the inverse Laplace of above equation

This equation can also be written as

## 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:

1. Delay Time.
2. Rise Time.
3. Peak Time.
4. Maximum Peak.
5. Settling Time.
6. 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

### 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.

### 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.

## Example - 1

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.

## Solution:

Given-

Rise Time:

Peak Time:

Settling Time:

Maximum overshoot:

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