Control Systems (CS)

Engr. Mehr Gul

Lecturer Electrical Engineering Deptt.

LectureTime Domain Analysis of 2nd Order Systems

Introduction

We have already discussed the affect of location of poles and zeroson the transient response of 1st order systems.

Compared to the simplicity of a first-order system, a second-ordersystem exhibits a wide range of responses that must be analyzedand described.

Varying a first-order system's parameter (T, K) simply changes thespeed and offset of the response

Whereas, changes in the parameters of a second-order system canchange the form of the response.

A second-order system can display characteristics much like a first-order system or, depending on component values, display dampedor pure oscillations for its transient response.

Introduction A general second-order system is characterized by

the following transfer function.

22

2

2 nn

n

sssR

sC

)(

)(

Introduction

un-damped natural frequency of the second order system,which is the frequency of oscillation of the system withoutdamping.

22

2

2 nn

n

sssR

sC

)(

)(

n

damping ratio of the second order system, which is a measureof the degree of resistance to change in the system output.

Example#1

42

42

sssR

sC

)(

)(

Determine the un-damped natural frequency and damping ratioof the following second order system.

42 n

22

2

2 nn

n

sssR

sC

)(

)(

Compare the numerator and denominator of the given transferfunction with the general 2nd order transfer function.

sec/radn 2 ssn 22

422 222 ssss nn 50.

1 n

Introduction

22

2

2 nn

n

sssR

sC

)(

)(

Two poles of the system are

1

1

2

2

nn

nn

Introduction

According the value of , a second-order system can be set intoone of the four categories:

1

1

2

2

nn

nn

1. Overdamped - when the system has two real distinct poles ( >1).

-a-b-c

j

Introduction

According the value of , a second-order system can be set intoone of the four categories:

1

1

2

2

nn

nn

2. Underdamped - when the system has two complex conjugate poles (0 <

Introduction

According the value of , a second-order system can be set intoone of the four categories:

1

1

2

2

nn

nn

3. Undamped - when the system has two imaginary poles ( = 0).

-a-b-c

j

Introduction

According the value of , a second-order system can be set intoone of the four categories:

1

1

2

2

nn

nn

4. Critically damped - when the system has two real but equal poles ( = 1).

-a-b-c

j

11

Example : Determine the un-damped natural frequency and dampingratio of the following second-order system.

Second Order System

93

9)(.1

2

sssG

168

16)(.2

2

sssG Solve them as your own

revision

END....