1. Time-domain vs Frequency -domain?
Frequency Response
Time-domain vs Frequency -domain?

2. Introduction
When a linear system is subjected to a sinusoidal input, the ultimate response (after a long time has passed) is also a sustained sinusoidal wave.
y(t)
m(t)
Input Wave
Output Wave Ay  Am
time
Question 1: What are the features of the output sinusoidal wave ?
Question 2: How do they change as a function of frequency ?

3. Frequency Response Analysis
y(t)
m(t)
Input Wave
Output Wave Ay  Am
time
AR= Ay / Am : Amplitude Ratio
 : Phase Shift (lag)

4. Computation of the Frequency Response
The frequency response, G ( j), is a Complex number.
Complex number
In Polar coordinates..
Real
Imaginary b a j s + =
Real
Imaginary f d

5. Computation of the Frequency Response
If g(s) is the transfer function of a process, g(j) represents its frequency response !!!

6. Bode-Nyquist Diagrams
The frequency response of a system is characterized by its
amplitude ratio,
phase angle
Frequency is a parameter.
Bode Diagram
(Logarithmic Plot)
Nyquist Diagram
(Polar Plot)

7. Bode Diagrams First-Order Process10 -2 -1 1 2 AR w
A Bode Diagram consist of a pair of plots showing:
How AR varies with frequency
Howvaries with frequency 10 -2 -1 1 2
-100
-50 f w
First-Order Process

8. A specific value of frequency  defines a point on this plot.
Nyquist Diagram
A Nyquist diagram (polar plot) is an alternative way to represent the frequency response.
Im g (j), ordinate
Re g (j), abscissa
A specific value of frequency  defines a point on this plot.

9. Nyquist Diagram
The Nyquist diagram contains the same information as the Bode diagram for the same system!!

10. Example - First-Order Process

11. Example - First-Order Process
Calculate AR andin terms of mod g(j))and arg (g(j))

12. Example - First-Order Process
Bode Diagram

13. Example - First-Order Process
Nyquist Diagram

14. Example - First-Order Process with Delay
Bode Diagram
With the addition of a delay term, AR remains the same, but the Phase Shift is significantly affected.

15. Example - First-Order Process
Nyquist Diagram
The net effect of the delay is to alter the phase characteristics of the process, which results in the circling of the origin at high frequencies with a decreasing radius.

16. Summary
Frequency response is a compact representation of process dynamics.
The frequency response is expressed through the amplitude ratio and the phase shift.
A Bode or a Nyquist Diagram can be used to describe the frequency response.
A first-order system has a maximum phase shift of 900