Time Response

Sources

1. Class Lecture

Terminologies

• Transient response is the system’s response after it changes from a steady state.
• Output response refers to the sum of the natural and forced response.
  • The forced response comes from the input after partial fraction expansion and inverse Laplace transform has been performed.
  • The natural response comes from the resulting partial fraction expansion and inverse Laplace transform performed on the transfer function, excluding the one from the input.
• Zeros are located at the numerator of a transfer function, and are represented by .
• Poles are located at the denominator of a transfer function, and are represented by .

The S Plane

• The axis represents , while the axis represents the

Finding the Evolution of the System Response

Steps:

1. Perform partial fraction expansion.
2. Solve for the inverse Laplace transform.

First Order Systems

If the input is a unit step, the general form of the first order system is

where

• is the output
• is the input
• is the transfer function

Therefore

(through derivations)

Characteristics of the Transient Response

Time Constant

When

As a result, is the time constant indicating the time needed for to decay to of its initial value.

Rise Time

The rise time refers to the time needed for the response to go from 10% to 90% of its steady state value.

Settling Time

The settling time refers to the time it takes for the response to reach and settle to 2% of its steady state value.

Second Order Systems

If the input is a unit step, the general form of the second order system is

TIP

You can easily find the roots of a second order system using a scientific calculator by clicking mode → EQN →

Types of Responses

Overdamped Response

Key	Value
Poles	Two real at and

Underdamped Response

Key	Value
Poles	Two complex at

NOTE

Undamped Response

Key	Value
Poles	Two imaginary at

Critically Damped Response

Key	Value
Poles	Two real at