Time-domain vs Frequency -domain? Frequency Response Time-domain vs Frequency -domain?

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 ?

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

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

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

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)

Bode Diagrams First-Order Process 10 -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

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.

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

Example - First-Order Process

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

Example - First-Order Process Bode Diagram

Example - First-Order Process Nyquist Diagram

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.

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.

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