Notes: Time Response Analysis | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


 
 
 
 
TIME RESPONSE ANALYSIS 
 
We can analyze the response of the control systems in both the time domain and the 
frequency domain. We will discuss frequency response analysis of control systems in later 
chapters. Let us now discuss about the time response analysis of control systems. 
What is Time Response? 
If the output of control system for an input  varies  with  respect  to  time,  then  it  is  
called the time response of the control system. The time response consists of two parts. 
 
 Transient response 
 Steady state response 
The response of control system in time domain is shown in the following figure. 
Page 2


 
 
 
 
TIME RESPONSE ANALYSIS 
 
We can analyze the response of the control systems in both the time domain and the 
frequency domain. We will discuss frequency response analysis of control systems in later 
chapters. Let us now discuss about the time response analysis of control systems. 
What is Time Response? 
If the output of control system for an input  varies  with  respect  to  time,  then  it  is  
called the time response of the control system. The time response consists of two parts. 
 
 Transient response 
 Steady state response 
The response of control system in time domain is shown in the following figure. 
 
 
 
 
 
Where, 
 
 c tr(t) is the transient response 
 c ss(t) is the steady state response 
Transient Response 
After applying input to the control system, output takes certain time to reach steady state. 
So, the output will be in transient state till it goes to a steady state. Therefore, the 
response of the control system during the transient state is known as transient response. 
The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ is infinity 
and practically, it is five times constant. 
Mathematically, we can write it as 
 
 
Steady state Response 
The part of the time response that remains even after the transient response has zero 
value for large values of ‘t’ is known as steady state response. This means, the transient 
response will be zero even during the steady state. 
Example 
Let us find the transient and steady state terms of the time response of the control system 
 
Here, the second term will be zero as t denotes infinity. So, this is the transient 
term. And the first term 10 remains even as t approaches infinity. So, this is the steady 
state term. 
Standard Test Signals 
The standard test signals are impulse, step, ramp and parabolic. These signals are used to 
know the performance of the control systems using time response of the output. 
Unit Impulse Signal 
A unit impulse signal, d(t) is defined as 
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TIME RESPONSE ANALYSIS 
 
We can analyze the response of the control systems in both the time domain and the 
frequency domain. We will discuss frequency response analysis of control systems in later 
chapters. Let us now discuss about the time response analysis of control systems. 
What is Time Response? 
If the output of control system for an input  varies  with  respect  to  time,  then  it  is  
called the time response of the control system. The time response consists of two parts. 
 
 Transient response 
 Steady state response 
The response of control system in time domain is shown in the following figure. 
 
 
 
 
 
Where, 
 
 c tr(t) is the transient response 
 c ss(t) is the steady state response 
Transient Response 
After applying input to the control system, output takes certain time to reach steady state. 
So, the output will be in transient state till it goes to a steady state. Therefore, the 
response of the control system during the transient state is known as transient response. 
The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ is infinity 
and practically, it is five times constant. 
Mathematically, we can write it as 
 
 
Steady state Response 
The part of the time response that remains even after the transient response has zero 
value for large values of ‘t’ is known as steady state response. This means, the transient 
response will be zero even during the steady state. 
Example 
Let us find the transient and steady state terms of the time response of the control system 
 
Here, the second term will be zero as t denotes infinity. So, this is the transient 
term. And the first term 10 remains even as t approaches infinity. So, this is the steady 
state term. 
Standard Test Signals 
The standard test signals are impulse, step, ramp and parabolic. These signals are used to 
know the performance of the control systems using time response of the output. 
Unit Impulse Signal 
A unit impulse signal, d(t) is defined as 
 
 
 
 
 
 
 
 
So, the unit impulse signal exists only at‘t’ is equal to zero. The area of this signal under 
small interval of time around‘t’ is equal to zero is one. The value of unit impulse signal is 
zero for all other values of‘t’. 
Unit Step Signal 
A unit step signal, u(t) is defined as 
 
 
Following figure shows unit step signal. 
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TIME RESPONSE ANALYSIS 
 
We can analyze the response of the control systems in both the time domain and the 
frequency domain. We will discuss frequency response analysis of control systems in later 
chapters. Let us now discuss about the time response analysis of control systems. 
What is Time Response? 
If the output of control system for an input  varies  with  respect  to  time,  then  it  is  
called the time response of the control system. The time response consists of two parts. 
 
 Transient response 
 Steady state response 
The response of control system in time domain is shown in the following figure. 
 
 
 
 
 
Where, 
 
 c tr(t) is the transient response 
 c ss(t) is the steady state response 
Transient Response 
After applying input to the control system, output takes certain time to reach steady state. 
So, the output will be in transient state till it goes to a steady state. Therefore, the 
response of the control system during the transient state is known as transient response. 
The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ is infinity 
and practically, it is five times constant. 
Mathematically, we can write it as 
 
 
Steady state Response 
The part of the time response that remains even after the transient response has zero 
value for large values of ‘t’ is known as steady state response. This means, the transient 
response will be zero even during the steady state. 
Example 
Let us find the transient and steady state terms of the time response of the control system 
 
Here, the second term will be zero as t denotes infinity. So, this is the transient 
term. And the first term 10 remains even as t approaches infinity. So, this is the steady 
state term. 
Standard Test Signals 
The standard test signals are impulse, step, ramp and parabolic. These signals are used to 
know the performance of the control systems using time response of the output. 
Unit Impulse Signal 
A unit impulse signal, d(t) is defined as 
 
 
 
 
 
 
 
 
So, the unit impulse signal exists only at‘t’ is equal to zero. The area of this signal under 
small interval of time around‘t’ is equal to zero is one. The value of unit impulse signal is 
zero for all other values of‘t’. 
Unit Step Signal 
A unit step signal, u(t) is defined as 
 
 
Following figure shows unit step signal. 
 
 
 
 
 
 
 
So, the unit step signal exists for all positive values of‘t’ including zero. And its value is one 
during this interval. The value of the unit step signal is zero for all negative values of‘t’. 
Unit Ramp Signal 
A unit ramp signal, r (t) is defined as 
 
 
 
So, the unit ramp signal exists for all positive values of‘t’ including zero. And its value 
increases linearly with respect to‘t’ during this interval. The value of unit ramp signal is 
zero for all negative values of‘t’. 
Unit Parabolic Signal 
A unit parabolic signal, p(t) is defined as, 
 
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TIME RESPONSE ANALYSIS 
 
We can analyze the response of the control systems in both the time domain and the 
frequency domain. We will discuss frequency response analysis of control systems in later 
chapters. Let us now discuss about the time response analysis of control systems. 
What is Time Response? 
If the output of control system for an input  varies  with  respect  to  time,  then  it  is  
called the time response of the control system. The time response consists of two parts. 
 
 Transient response 
 Steady state response 
The response of control system in time domain is shown in the following figure. 
 
 
 
 
 
Where, 
 
 c tr(t) is the transient response 
 c ss(t) is the steady state response 
Transient Response 
After applying input to the control system, output takes certain time to reach steady state. 
So, the output will be in transient state till it goes to a steady state. Therefore, the 
response of the control system during the transient state is known as transient response. 
The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ is infinity 
and practically, it is five times constant. 
Mathematically, we can write it as 
 
 
Steady state Response 
The part of the time response that remains even after the transient response has zero 
value for large values of ‘t’ is known as steady state response. This means, the transient 
response will be zero even during the steady state. 
Example 
Let us find the transient and steady state terms of the time response of the control system 
 
Here, the second term will be zero as t denotes infinity. So, this is the transient 
term. And the first term 10 remains even as t approaches infinity. So, this is the steady 
state term. 
Standard Test Signals 
The standard test signals are impulse, step, ramp and parabolic. These signals are used to 
know the performance of the control systems using time response of the output. 
Unit Impulse Signal 
A unit impulse signal, d(t) is defined as 
 
 
 
 
 
 
 
 
So, the unit impulse signal exists only at‘t’ is equal to zero. The area of this signal under 
small interval of time around‘t’ is equal to zero is one. The value of unit impulse signal is 
zero for all other values of‘t’. 
Unit Step Signal 
A unit step signal, u(t) is defined as 
 
 
Following figure shows unit step signal. 
 
 
 
 
 
 
 
So, the unit step signal exists for all positive values of‘t’ including zero. And its value is one 
during this interval. The value of the unit step signal is zero for all negative values of‘t’. 
Unit Ramp Signal 
A unit ramp signal, r (t) is defined as 
 
 
 
So, the unit ramp signal exists for all positive values of‘t’ including zero. And its value 
increases linearly with respect to‘t’ during this interval. The value of unit ramp signal is 
zero for all negative values of‘t’. 
Unit Parabolic Signal 
A unit parabolic signal, p(t) is defined as, 
 
 
 
 
 
 
 
 
So, the unit parabolic signal exists for all the positive values of‘t’ including zero. And its 
value increases non-linearly with respect to‘t’ during this interval. The value of the unit 
parabolic signal is zero for all the negative values of‘t’. 
In this chapter, let us discuss the time response of the first order system. Consider the 
following block diagram of the closed loop control system. Here, an open loop transfer 
function, 1/sT is connected with a unity negative feedback.