Frequency Response Methods

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Frequency Response Methods

• Modern Control Systems
• Lecture 18

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Outline

• What is frequency response
• Frequency response plots
• Bode diagrams of basic factors

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What is frequency response

• So far we have described the response and
  performance of a system in terms of complex
  frequency variable ssj? and the location of
  poles and zeros in the s-plane. An important
  alternative approach to system analysis and
  design is the frequency response method.
• The frequency response of a system is defined as
  the steady-state response of the system to a
  sinusoidal input signal. We will investigate the
  steady-state response of the system to the
  sinusoidal input as the frequency varies.

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When the input signal is a sinusoid, the
resulting output signal for LTI systems is
sinusoidal in the steady state, it differs from
the input only in amplitude and phase.
where p1, p2,,pn are distinctive poles,
then in partial fraction expansion form, we have
Taking the inverse Laplace transform yields
Suppose the system is stable, then all the poles
are located in the left half plane and thus the
exponential terms decay to zero as t?8. Hence,
the steady-state response of the system is
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That is, the steady-state response depends only
on the magnitude and phase of T(j?).

• Advantages of the frequency response method
• The sinusoidal input signal for various ranges of
  frequency and amplitude is readily available.
• It is the most reliable and uncomplicated method
  for the experimental analysis of a system.
• Control of system bandwidth.
• The TF describing the sinusoidal steady-state
  behaviour of the system is easily obtained by
  replacing s with j? in the system TF.

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Frequency response plots

• Polar plot
• The TF G(s) can be described in the frequency
  domain by

The above equation is used for the polar plot
representation of the frequency response in the
polar plane.
polar plane
Alternatively, the TF G(j?) can be represented by
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Frequency response plots (contd)
Example polar plot
Consider a simple RC circuit. The TF of the
system is
The sinusoidal steady-state TF is
The polar plot is obtained from
polar plot
To draw the polar plot, R(?) and X(?) at typical
frequencies, e.g., ?0, 8, are to be determined.
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Frequency response plots (contd)

• Limitations of polar plots
• The addition of poles and zeros requires the
  recalculation of the frequency response.
• The effect of individual poles and zeros is not
  indicated.

A more widely used graphical tool to plot
frequency response is the Bode diagram.

• Bode plot

The TF in the frequency domain can be written as
For a Bode diagram, we normally use
Magnitude versus ? and phase versus ? are plotted
separately.
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Bode diagram

• Advantages of Bode plots
• Multiplication of magnitudes can be converted
  into addition by virtue of the definition of
  logarithmic gain.
• Straight-line asymptotes are simple to be used
  for sketching an approximate log-magnitude curve.

The use of a logarithmic scale for the frequency
is a more judicious choice than a linear scale of
frequency as this expands the low frequency
range, which is more important in practical
systems.
An interval of two frequencies with a ratio equal
to 10 is called a decade. The slope of the
asymptotic line in the figure is -20dB/decade.
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Bode diagram of constant gain K

• For a generalised TF,

• there often exist four basic factors, i.e.,
• Constant gain K
• Poles (or zeros) at the origin (j?)
• Poles (or zeros) on the real axis (j?t1)
• Complex conjugate poles (or zeros)
  1(2?/?n)j?(j?/?n)2

We will determine the Bode diagram for each of
these factors.
Bode diagram of constant gain K
The logarithmic gain is a horizontal line on the
Bode diagram,
and the phase angle is
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Bode diagram of poles (or zeros) at the origin
(j?)

• The log magnitude of a pole at the origin is

The slope of the magnitude curve is -20dB/decade
for the pole at the origin. For a multiple pole
at the origin, the log magnitude is
Bode diagram of (j?)N
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Bode diagram of poles (or zeros) on the real axis

• The log magnitude of (1j?t)-1 is

The asymptotic curve for ?ltlt1/t is -10log10 dB,
and the asymptotic curve for ?gtgt1/t is -20log?t
with a slope of -20dB/decade. The two asymptotes
intersect at the 0dB line when ?1/t, the break
frequency or corner frequency. The log magnitude
at ?1/t is -10log2-3dB.
The phase angle of (1j?t)-1 is
Bode diagram of (1j?t)-1
The Bode diagram of (1j?t) is obtained in the
same manner as that of (1j?t)-1.
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Bode diagram of complex conjugate poles (or zeros)

• The normalised form of a pair of complex
  conjugate poles can be written as

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Bode diagram of complex conjugate poles (or
zeros) (contd)
The magnitude asymptotes intersect at the 0dB
line when u?/?n1.
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Bode diagram of complex conjugate poles (or
zeros) (contd)
Bode diagram of (1(2?/?n)j?(j?/?n)2-1
The difference between the actual magnitude curve
and the asymptotic approximation is a function of
?. The max value of the frequency response
occurs at the resonant frequency, ?r.
When ??0, ?r? ?n.
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Bode diagram of complex conjugate poles (or
zeros) (contd)
The resonant frequency ?r is determined by taking
the derivative of the magnitude of
with respect to u, and setting it equal to zero.
The resonant frequency is
and the max value of the magnitude of G(?) is