The document discusses the response of first and second order systems. It defines poles and zeros as values that make the transfer function infinite or zero. For first order systems, it describes the transient response as gradual change from initial to final output. The time constant determines how quickly the system responds. For second order systems, the natural frequency and damping ratio characterize the response. Underdamped systems oscillate with declining amplitude, with specifications like rise time and settling time defined.

1. CHAPTER 5
THE RESPONSE TIME
ANALYSIS

2. 5.1 FIRST AND SECOND ORDER
SYSTEM
INTRODUCTION

3. • The concept of poles and zeros, fundamental to the
analysis of and design of control system, simplifies the
evaluation of system response.
• The poles of a transfer function are:
i. Values of the Laplace Transform variables s, that cause
the transfer function to become infinite.
ii. Any roots of the denominator of the transfer function
that are common to roots of the numerator.
• The zeros of a transfer function are:
i. The values of the Laplace Transform variable s, that
cause the transfer function to become zero.
ii. Any roots of the numerator of the transfer function that
are common to roots of the denominator.

4. • The concept of poles and zeros, fundamental to the analysis
of and design of control system, simplifies the evaluation of
system response.
• The poles of a transfer function are:
i. Values of the Laplace Transform variables s, that cause the
transfer function to become infinite.
ii. Any roots of the denominator of the transfer function that are
common to roots of the numerator.
• The zeros of a transfer function are:
i. The values of the Laplace Transform variable s, that cause
the transfer function to become zero.
ii. Any roots of the numerator of the transfer function that are
common to roots of the denominator.

5. • The output response of a system is a sum of
i. Forced response
ii. Natural response
a) System showing an input and an output
b) Pole-zero plot of the system

6. c) Evolution of a system response. Follow the blue
arrows to see the evolution of system
component generated by the pole or zero

7. Effect of a real-axis pole upon transient response
a) First-order system
b) Pole plot of the system

8. • General form:
C (s) K
G (s) = =
R( s ) τs + 1
• Problem: Derive the transfer function for the following circuit
1
G (s) =
RCs + 1

9. • Transient Response: Gradual change of output from initial to
the desired condition.
• Block diagram representation:
K Where,
R(s) C(s) K : Gain
τs + 1 τ : Time constant
• By definition itself, the input to the system should be a step
function which is given by the following:
1
R( s) =
s

10. • General form:
C (s) K
G (s) = = C ( s) = G ( s) R( s)
R( s ) τs + 1
• Output response:
 1  K 
C ( s ) =   
 s  τs + 1 
A B
= +
s τs + 1
B −t τ
c(t ) = A + e
τ

11. • Problem: Find the forced and natural responses for the
following systems

12. First-order system response to a unit step

13. • Time constant, τ
– The time for e-at to decay 37% of its 1
τ=
initial value. a
• Rise time, tr
– The time for the waveform to go 2. 2
tr =
from 0.1 to 0.9 of its final value. a
• Settling time, ts
– The time for the response to reach, 4
ts =
and stay within 2% of its final value. a

14. • Problem: For a system with the transfer function shown
below, find the relevant response specifications
50
G(s) =
s + 50
i. Time constant, τ
ii. Settling time, ts
iii. Rise time, tr

15. • General form:
Kω n
2
G( s ) = 2
s + 2ςωn s + ωn
2
Where,
K : Gain
ς : Damping ratio
ωn : Undamped natural frequency
• Roots of denominator:
s 2 + 2ςωn s + ωn = 0
2
s1, 2 = −ςωn ± ωn ς 2 − 1

16. • Natural frequency, ωn
– Frequency of oscillation of the system without damping.
• Damping ratio, ς
– Quantity that compares the exponential decay frequency of
the envelope to the natural frequency.
Exponential decay frequency
ς=
Natural frequency (rad/s)

17. • Problem: Find the step response for the following transfer
function
225
G( s ) = 2
s + 30 s + 225
• Answer:
c( t ) = 1 − e −15t − 15te −15t

18. • Problem: For each of the transfer function, find the values of
ς and ωn, as well as characterize the nature of the response.
400
a) G ( s ) =
s 2 + 12s + 400
b) G ( s ) = 2 900
s + 90 s + 900
c) G ( s ) = 225
s 2 + 30 s + 225
d) 625
G( s ) = 2
s + 625

21. • Step responses for second-order system damping cases

22. • Pole plot for the underdamped second-order system

23. • Second-order response as a function of damping ratio

24. • Second-order response as a function of damping ratio

25. • When 0 < ς < 1, the transfer function is given by the
following.
Kω n 2 Where,
G( s ) = ωd = ωn 1 − ς 2
( s + ςωn + jωd )( s + ςωn − jωd )
• Pole position:

26. • Second-order response components generated by complex
poles

27. • Second-order underdamped responses for damping ratio
value

28. • Second-order underdamped response specifications

29. • Rise time, Tr
– The time for the waveform to go from 0.1 to 0.9 of its
final value.
• Peak time, Tp π
Tp = 2
– The time required to reach the first ωn 1 − ζ
or maximum peak.
• Settling time, Ts
4
– The time required for the transient’s Ts =
ζω n
damped oscillation to reach and stay
within ±2% of the steady-state value.

30. • Percent overshoot, %OS
– The amount that the waveform overshoots the steady-state, or
final value at peak time, expressed as a percentage of the
steady-state value.
− (ζπ / 1−ζ 2 )
%OS = e × 100%
− ln(%OS / 100)
ζ =
π 2 + ln 2 (%OS / 100)

31. • Percent overshoot versus damping ratio

32. • Lines of constant peak time Tp, settling time Ts and percent
overshoot %OS
Ts2 < Ts1
Tp2 < Tp1
%OS1 < %OS2

33. • Step responses of second-order underdamped systems as
poles move
a) With constant
real part
b) With constant
imaginary part

34. • Step responses of second-order underdamped systems as
poles move
c) With constant damping ratio