1. Nyquist (1) Hany Ferdinando Dept. of Electrical Engineering
Petra Christian University

2. Nyquist (1) - Hany Ferdinando
General Overview
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

3. Nyquist (1) - Hany Ferdinando
What?
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
Nyquist (1) - Hany Ferdinando

4. Nyquist (1) - Hany Ferdinando
Polar Coordinate?
There are two parameters:
Radius, measured from the origin
Angle (from positive real axis)
Positive angle is counter clock wise
Negative angle is clock wise
For reference, students can study the general concept for algebra and geometry
Nyquist (1) - Hany Ferdinando

5. Advantage and Disadvantage
It shows the frequency response characteristic of a system over the entire freq. range in a single plot
Disadvantage:
It does not clearly indicate the contribution of each individual factor of the open-loop transfer function
Nyquist (1) - Hany Ferdinando

6. Nyquist (1) - Hany Ferdinando
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!!!
Nyquist (1) - Hany Ferdinando

7. Derivative/integral factors (jw)±1
Polar plot of (jw)-1 is negative imaginary axis
Polar plot of (jw) is positive imaginary axis
Nyquist (1) - Hany Ferdinando

8. 1st-order factors (1+jw)±1
For w = 0  1 angle 0o
For w = 1/T  1/√2 angle -45o
For w = ∞  0 angle -90o
Nyquist (1) - Hany Ferdinando

9. 1st-order factors (1+jw)±1
Nyquist (1) - Hany Ferdinando

10. 1st-order factors (1+jw)±1
For w = 0  1 angle 0o
For w = 1/T  √2 angle 45o
For w = ∞  ∞ angle 90o
Nyquist (1) - Hany Ferdinando

11. 1st-order factors (1+jw)±1
Nyquist (1) - Hany Ferdinando

12. Nyquist (1) - Hany Ferdinando
Quadratic Factor
[1+2z(jw/wn)+(jw/wn)2]-1
For w0, G(jw) = 1 angle 0o
For w∞, G(jw) = 0 angle -180o
Nyquist (1) - Hany Ferdinando

13. Nyquist (1) - Hany Ferdinando
Quadratic Factor
Nyquist (1) - Hany Ferdinando

14. Nyquist (1) - Hany Ferdinando
Quadratic Factor
[1+2z(jw/wn)+(jw/wn)2]
For w0, G(jw) = 1 angle 0o
For w∞, G(jw) = ∞ angle 180o
Nyquist (1) - Hany Ferdinando

15. Nyquist (1) - Hany Ferdinando
Quadratic Factor
Nyquist (1) - Hany Ferdinando

16. 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
Nyquist (1) - Hany Ferdinando

17. 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
Nyquist (1) - Hany Ferdinando

18. 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
Nyquist (1) - Hany Ferdinando

19. Nyquist (1) - Hany Ferdinando
Polar plot of type 0, 1 and 2
w = 0 ∞ w
Type 0
Type 1
Type 2
Nyquist (1) - Hany Ferdinando

20. Polar Plot of high freq. range
n-m = 1
Nyquist (1) - Hany Ferdinando

21. Nyquist (1) - Hany Ferdinando
Nyquist in Matlab
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
Nyquist (1) - Hany Ferdinando

22. Nyquist (1) - Hany Ferdinando
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.
Nyquist (1) - Hany Ferdinando