Transcript Nyquist (1) - PCU Teaching Staffs

Nyquist (1)
Hany Ferdinando
Dept. of Electrical Engineering
Petra Christian University
General Overview

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Relation between Magnitude and
Phase Angle
Freq. response can be analyzed also
via Polar Plot (the polar plot is also
called as Nyquist plot)
Advantages and disadvantages
How to plot Nyquist using Matlab is
discussed here
Nyquist (1) - Hany Ferdinando
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What?
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Bode diagram uses two plots to show
the frequency response of plants
Nyquist combines both plots into a
single plot in polar coordinate as w is
varied from 0 to ∞One must have
knowledge in using complex number
both in general notation and in
phasor form
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Polar Coordinate?

There are two parameters:
• Radius, measured from the origin
• Angle (from positive real axis)
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Positive angle is counter clock wise
Negative angle is clock wise
For reference, students can study the
general concept for algebra and
geometry
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Advantage and Disadvantage

Advantage:
• It shows the frequency response
characteristic of a system over the
entire freq. range in a single plot
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Disadvantage:
• It does not clearly indicate the
contribution of each individual factor of
the open-loop transfer function
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How to plot it?
The idea is presented in:
 Integral/derivative factor
 First order factor
 Quadratic factor
Students have to exercise themselves how
to plot the nyquist plot for other factors!!!
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Derivative/integral factors (jw)±1
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Polar plot of (jw)-1 is negative
imaginary axis
Polar plot of (jw) is positive
imaginary axis
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1st-order factors (1+jw)±1

(1+jw)-1
•
1
1
G( jw ) 

, angle   tan 1 wT
1  jwT
1  w 2T 2
• For w = 0  1 angle 0o
• For w = 1/T  1/√2 angle -45o
• For w = ∞  0 angle -90o
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1st-order factors (1+jw)±1
Nyquist Diagram
1
0.8
0.6
Imaginary Axis
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Real Axis
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1st-order factors (1+jw)±1

(1+jw)-1
• G( jw )  1  jwT  1  w 2T 2 , angle  tan 1 wT
• For w = 0  1 angle 0o
• For w = 1/T  √2 angle 45o
• For w = ∞  ∞ angle 90o
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1st-order factors (1+jw)±1
Nyquist Diagram
5
4
3
Imaginary Axis
2
1
0
-1
-2
-3
-4
-5
0
0.5
1
1.5
Real Axis
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Quadratic Factor

[1+2z(jw/wn)+(jw/wn)2]-1
• G( jw ) 
1
 w
1  2z  j
 wn
  w
   j
  wn



2
• For w0, G(jw) = 1 angle 0o
• For w∞, G(jw) = 0 angle -180o
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Quadratic Factor
Nyquist Diagram
1.5
1
Imaginary Axis
0.5
0
-0.5
-1
-1.5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Real Axis
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Quadratic Factor

[1+2z(jw/wn)+(jw/wn)2]
•
 w   w 
   j

G ( jw )  1  2z  j
 wn   wn 
2
• For w0, G(jw) = 1 angle 0o
• For w∞, G(jw) = ∞ angle 180o
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Quadratic Factor
Nyquist Diagram
20
15
10
Imaginary Axis
5
0
-5
-10
-15
-20
-250
-200
-150
-100
-50
0
50
Real Axis
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General Shapes of Polar Plot

Type 0 systems: the starting point
(w=0) is finite on positive real axis.
The tangent to polar plot at w=0 is
perpendicular to the real axis. The
terminal point (w=∞) is at the origin
and the curve is tangent to one of
the axes
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General Shapes of Polar Plot

Type 1 systems: at w=0, the
magnitude is infinity and phase angle
is -90o. At w=∞, the magnitude is
zero and the curve converges to the
origin and is tangent to one of the
axes
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General Shapes of Polar Plot

Type 2 systems: at w=0, the
magnitude is infinity and the phase
angle is -180o. At w=∞ the
magnitude becomes zero and the
curve is tangent to one of the axes
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Polar plot of type 0, 1 and 2
w
Type 2
∞
w
0
w
∞
∞
w
Type 1
w=0
Type 0
w
0
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Polar Plot of high freq. range
n-m = 1
n-m = 1
n-m = 1
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Nyquist in Matlab
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Use [re,im,w]=nyquist(sys)
‘sys’ may be filled with (num,den) or
transfer function or (A,B,C,D) in
state space
In Nyquist, it is important to see the
direction of the curve
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Next…
The Nyquist plot is already discussed, the subjects
are to plot several general equation in Nyquist and
how to use Matlab to plot nyquist.
The next class is Nyquist stability criterion and
phase-gain margin. Students have to prepare
themselves for this topic.
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