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 $0$.
• Poles are located at the denominator of a transfer function, and are represented by $X$.

The S Plane

• The $y$ axis represents $J\omega$, while the $x$ axis represents the $\varphi$

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

$$C(s)=R(s)G(s)=\frac{a}{s(s+a)}$$

where

• $C(s)$ is the output
• $R(s)$ is the input
• $G(s)$ is the transfer function

Therefore

$$c(t)=\text{forced response }+\text{natural response}=1-e^{-at}$$

(through derivations)

Characteristics of the Transient Response

Time Constant

When $t=\frac{1}{a}$

$$\begin{array}{rlr}{e}^{-at}{|}_{t=\frac{1}{a}}& =e^{-1}& =0.37\\ c(t)& =1-0.37& =0.63\end{array}$$

As a result, $\frac{1}{a}$ is the time constant indicating the time needed for $e^{-at}$ to decay to $37\%$ of its initial value.

Rise Time

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

$T_r=\frac{2.2}{a}$

Settling Time

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

$$T_s=\frac{4}{a}$$

Second Order Systems

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

$$G(s)=\frac{b}{s^2+as+b}$$

TIP

You can easily find the roots of a second order system using a scientific calculator by clicking mode -> EQN -> $a{X}^2+bX+C=0$

Types of Responses

Overdamped Response

Key	Value
Poles	Two real at $-{\sigma }_1$ and $-{\sigma }_2$
$c(t)$	$K_1+K_2{e}^{-{\sigma }_{1}t}+K_3{e}^{-{\sigma }_{2}t}$

Underdamped Response

Key	Value
Poles	Two complex at $-{\sigma }_d\pm j{\omega }_1$
$c(t)$	$K_1+K_4{e}^{-{\sigma }_d}\cos ({\omega }_{1}t+\varphi )$

NOTE

• $\varphi ={\tan }^{-1}\frac{K_3}{K_2}$
• $K_4=\sqrt{K_2^2+K_3^2}$

Undamped Response

Key	Value
Poles	Two imaginary at $at\pm j{\omega }_1$
$c(t)$	$K_1+K_4\cos (3t+\varphi )$

Critically Damped Response

Key	Value
Poles	Two real at $-{\sigma }_1$
$c(t)$	$K_1+K_2{e}^{-{\sigma }_{1}t}+K_{3}t{e}^{-{\sigma }_{2}t}$