Frequency Domain techniques

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Frequency Domain techniques

• Stability in the Frequency Domain

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Structure

• Mapping Contours in the s-plane
• Nyquist Criterion
• Gain Phase Margin

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Mapping Contours in the s-plane
We are interested in mapping the contours in the
s-plane by a function F(s), where, F(s)1L(s)0
That is, the characteristic equation. A contour
map is a contour or trajectory in one plane
mapped or translated into another plane by a
relation F(s).
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Mapping Contours in the s-plane
Since s is a complex variable, i.e. s?j?, then
F(s) is itself complex. So we can represent
F(s)ujv and can be represented on the complex
plane with co-ordinates u and v. E.g if
F(s)2s1, consider mapping the unit square
contour in the s-plane to the complex plane.
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Mapping Contours in the s-plane
Now if we consider the points A,B,C and D in the
s-plane contour map, they map to the points A,B,C
and D shown in the F(s) plane. This type of
mapping is termed a conformal mapping. Note also
a closed contour in the s-plane results in a
closed contour in F(s)-plane
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Mapping Contours in the s-plane
Consider a function F(s)s/(s2), what is the
mapping of the unit square contour in the
F(s)-plane? It is as below, calculated by
calculating ujv for each pint in the s-plane,
e.g. A in the s-plane is 1j1, and in the
F(s)-plane (42j)/10. Similarly 1 in the s-plane
equates to a point 1/3 in the F(s)-plane.
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Cauchys Theorem
If a contour Ts in the s-plane encircles Z zeros
and P poles and does not pass through any poles
or zeros of F(s) and the transversal is in the
clockwise direction along the contour, the
corresponding contour TF in the F(s) plane
encircles the origin of the F(s)-plane NZ-P
times in the clockwise direction. So for the two
cases considered previously, the F(s)-plane
encircles the origin once because
NZ-P1. Cauchys theorem is best understood by
thinking of F(s) in terms of the angle due to
each pole and zero as the contour Ts is traversed
in a clockwise direction.
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A Further example
Consider F(s)s/(s1/2) and the contour shown
below note in this case NZ-P0, so in the
F(s)-plane the contour does not encircle the
origin.
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Cauchys Theorem What does it mean
Cauchys theorem is best understood by thinking
of F(s) in terms of the angle due to each pole
and zero as the contour Ts is traversed in a
clockwise direction. Consider F(s)(sz1)
(sz2)/ (sp1) (sp2) Recall we can rewrite
this as F(s)?F(s)sz1 sz2/ sp1
sp2 ?F(s)?(sz1) ?(sz2)- ?(sp1)-
?(sp2) Alternatively, F(s)(?z1 ?z2-?p1
-?p2)
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Cauchys Theorem
Now if we consider the vectors for specific
contour, we can determine the angles a s
traverses the contour. Clearly the net angle
changes as s traverses the contour along a full
rotation of 360?.
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Nyquist Criterion

• For stability all the roots (zeros) of F(s) must
  lie in the LHS of the s-plane.
• So in the s-plane we choose a contour that
  encloses the whole RHS of the s-plane and we
  determine if any zeros of F(s) lie within the
  contour by using Cauchys theorem.
• i.e. we plot a contour in the F(s) plane
  corresponding to the above contour and look for
  any encirclements of the origin.

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The Nyquist Contour
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A Change of Function.
Note the Nyquist criterion is concerned with the
c.e. F(s)1L(s) And the number of
encirclements of the origin of the F(s)-plane. If
we rewrite this as F(s)F(s)-1L(s), Then
L(s) is normally available in factorised form and
then we are interested in the number of
encirclements of the point 1 as opposed to the
origin.
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The Nyquist Criterion
A feedback system is stable if and only if, for
the contour TL, the number of counter-clockwise
encirclements of the (-1,0) point is equal to the
number of poles of L(s) with positive real parts.
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System with a pole at the origin
Consider a single loop control system,
GH(s)K/(s(s?1). Recall the condition of
Cauchys theorem that states that the contour
cannot pass through a pole at the origin.
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System with a pole at the origin

• At the origin the small detour is obtained by
  setting s?ej ? , where ? is a very small number,
  and allowing ? to vary from -90? at ?0- to 90?
  at ?0 , passing through 0? at ?0.
• From ?0 to ? the function GH(s) is mapped as a
  real frequency polar plot.
• From ? ? to -? is mapped into the point zero at
  the origin.
• From From ?0- to -? is mapped by setting s-j ?
  in GH(s)

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And Generally.

• The plot in the range -?lt ?lt0- is the complex
  conjugate of 0 lt ?lt ? , and the polar plot will
  be symmetrical about the u-axis.
• The magnitude of GH(s) as srej? and r-gt ? will
  normally approach zero or a constant.

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System with three poles.
Consider a single loop system, GH(s)K/s(sT1)
(sT2), the Nyquist contour is shown below,
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So how did we get there?

• Note that the mapping is symmetrical for /-
• The origin of the s-plane maps into a semi circle
  of infinite radius
• The semi-circle in the s-plane maps onto GH(s)0
• So we need only plot the portion of the contour
  that is the real frequency polar plot for 0 lt ?lt
  ?

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So we could go unstable
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Gain and Phase Margin

• The gain margin is the increase in the system
  gain when the phase is -180? that will result in
  a marginally stable system with the intersection
  of 1j0 point on the Nyquist diagram.
• The phase margin is the amount of phase shift of
  the GH(j?) at unity magnitude that will result in
  a marginally stable system with the intersection
  of the 1j0 point on the Nyquist diagram.

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Gain and Phase Margin

• The gain and phase margins are readily obtained
  from the Bode diagram.
• The crtitical point for stability is u-1, v0 in
  the GH(j?) plane which is equivalent to
• Logaritmic magnitude fo 0dB
• Phase angle of -180 ?

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An example.
Gain margin 15dB, Phase margin43 ?
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Phase margin and damping factor

• Consider a general 2nd order system then we can
  show that the phase margin is related to the
  damping factor by the following
• ?0.01?PM
• This is reasonably accurate for ?lt0.7 and is
  useful for correlating time-domain response to
  frequency response.

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Recall the example
In the example we used the phase margin was 43
?, So the damping factor is approximately 0.43.
So the percentage overshoot, P.O. is 22.