Frequency Response Analysis

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Lecture 18

• Frequency Response Analysis

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Frequency Response
u(t)Asin(wt)
YAsin (wtf)
G
(s)
p
Force dynamic process with sin ?t ,
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Perturbation of a first order system

• Consider a first order system
• Let u(t) be a sinusoidal input
• The Laplace transform is
• Hence,

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Perturbation (continued)

• Expand and take inverse Laplace transforms
• At steady-state
• This can be transformed into
• where

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Input Output
is the output amplitude
is the amplitude ratio
(AR) ? is the phase angle, phase shift. AR and ?
are functions of ?. For a general transfer
function
Chapter 13
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Shortcut method for finding the frequency
response
Example 1 find the frequency response of 1st
order system
Chapter 13
K1 K2
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General formula to determine AR and phase
shift (Eq 13-2213-24)
Chapter 13
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Example 13.2 Calculate the amplitude ratio and
phase angle for overdamped 2nd order transfer
function
Using Eq 13-24
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Bode diagram for a first-order process
AR (or ARN) and ø are each plotted as a function
of ? (or ?t)
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Bode Plot of Complex Transfer Functions
Bode plot provides a convenient display of the
frequency response characteristics of a transfer
function model. It consists of plots of AR and f
as a function of ?.

• Break transfer function into a product of simple
  transfer functions.
• Identify AR(w) and f(w) of each simple transfer
  function.
• Combine to get Ar(w) and f(w) for complex
  transfer function according to properties.
• Plot results as a function of w.

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Bode plot for Second-Order Process
Substituting s j? and rearranging yields
For underdamped systems, the amplitude ratio plot
exhibits a maximum (for values of 0lt?lt )
at the resonant frequency
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Bode plot for Second-Order Process
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Frequency Response Characteristics of Controllers
(Table 13.5)
Proportional-Integral Controller. A
proportional-integral (PI) controller has the
transfer function,
Substitute s j?
Thus, the amplitude ratio and phase angle are
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For the ideal proportional-derivative (PD)
controller
Parallel PID Controller.
Series PID Controller with a Derivative Filter.
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Bode plots of ideal parallel PID controller and
series PID controller with derivative filter (a
0.1). Idea parallel Series with Derivative
Filter A derivative filter can bound the AR
at high frequency region.