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

Nyquist (1)Nyquist (1)

Hany FerdinandoHany FerdinandoDept. of Electrical EngineeringDept. of Electrical Engineering

Petra Christian UniversityPetra Christian University

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

General OverviewGeneral Overview

Relation between Magnitude and Relation between Magnitude and Phase AnglePhase Angle

Freq. response can be analyzed also Freq. response can be analyzed also via Polar Plot (the polar plot is also via Polar Plot (the polar plot is also called as Nyquist plot)called as Nyquist plot)

Advantages and disadvantagesAdvantages and disadvantages How to plot Nyquist using Matlab is How to plot Nyquist using Matlab is

discussed herediscussed here


What?What?

Bode diagram uses two plots to show Bode diagram uses two plots to show the frequency response of plantsthe frequency response of plants

Nyquist combines both plots into a Nyquist combines both plots into a single plot in polar coordinate as single plot in polar coordinate as is is varied from 0 to ∞One must have varied from 0 to ∞One must have knowledge in using complex number knowledge in using complex number both in general notation and in both in general notation and in phasor formphasor form


Polar Coordinate?Polar Coordinate?

There are two parameters:There are two parameters:• Radius, measured from the originRadius, measured from the origin• Angle (from positive real axis)Angle (from positive real axis)

Positive angle is counter clock wisePositive angle is counter clock wise Negative angle is clock wiseNegative angle is clock wise

For reference, students can study the For reference, students can study the general concept for algebra and general concept for algebra and geometrygeometry


Advantage and DisadvantageAdvantage and Disadvantage

Advantage:Advantage:• It shows the frequency response It shows the frequency response

characteristic of a system over the characteristic of a system over the entire freq. range in a single plotentire freq. range in a single plot

Disadvantage:Disadvantage:• It does not clearly indicate the It does not clearly indicate the

contribution of each individual factor of contribution of each individual factor of the open-loop transfer functionthe open-loop transfer function


How to plot it?How to plot it?

The idea is presented in:The idea is presented in: Integral/derivative factorIntegral/derivative factor First order factorFirst order factor Quadratic factorQuadratic factor

Students have to exercise themselves how Students have to exercise themselves how to plot the nyquist plot for other factors!!!to plot the nyquist plot for other factors!!!


Derivative/integral factors (jDerivative/integral factors (j))±±11

Polar plot of (jPolar plot of (j))--11 is negative is negative imaginary axisimaginary axis

Polar plot of (jPolar plot of (j) is positive imaginary ) is positive imaginary axisaxis


11stst-order factors (1+j-order factors (1+j))±±11

(1+j(1+j))--11

•

• For For = 0 = 0 1 angle 0 1 angle 0oo

• For For = 1/T = 1/T 1/ 1/√2 angle -45√2 angle -45oo

• For For = ∞ = ∞ 0 angle -90 0 angle -90oo

TangleTTj

jG

1

22tan,

1

1

1

1)(



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Nyquist Diagram

Real Axis

Imag

inar

y A

xis



(1+j(1+j))--11

• • For For = 0 = 0 1 angle 0 1 angle 0oo

• For For = 1/T = 1/T √2 angle 45√2 angle 45oo

• For For = ∞ = ∞ ∞ angle 90 ∞ angle 90oo

TangleTTjjG 122 tan,11)(



0 0.5 1 1.5-5

-4

-3

-2

-1

0

1

2

3

4

5Nyquist Diagram

Real Axis

Imag

inar

y A

xis


Quadratic FactorQuadratic Factor

[1+2[1+2(j(j//nn)+(j)+(j//nn))22]]-1-1

•

• For For 0, G(j0, G(j) = 1 angle 0) = 1 angle 0oo

• For For ∞, G(j∞, G(j) = 0 angle -180) = 0 angle -180oo

2

21

1)(

nn

jj

jG



-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2-1.5

-1

-0.5

0

0.5

1

1.5Nyquist Diagram

Real Axis

Imag

inar

y A

xis



[1+2[1+2(j(j//nn)+(j)+(j//nn))22]]

•

• For For 0, G(j0, G(j) = 1 angle 0) = 1 angle 0oo

• For For ∞, G(j∞, G(j) = ∞ angle 180) = ∞ angle 180oo

2

21)(

nn

jjjG



-250 -200 -150 -100 -50 0 50-20

-15

-10

-5

0

5

10

15

20Nyquist Diagram

Real Axis

Imag

inar

y A

xis


General Shapes of Polar PlotGeneral Shapes of Polar Plot

Type 0 systems: Type 0 systems: the starting pointthe starting point ((=0) is finite on positive real axis. =0) is finite on positive real axis. The tangent to polar plot at The tangent to polar plot at =0 is =0 is perpendicular to the real axis. perpendicular to the real axis. The The terminal pointterminal point ( (=∞) is at the origin =∞) is at the origin and the curve is tangent to one of and the curve is tangent to one of the axesthe axes



Type 1 systems: at Type 1 systems: at =0, the =0, the magnitude is infinity and phase angle magnitude is infinity and phase angle is -90is -90oo. At . At =∞, the magnitude is =∞, the magnitude is zero and the curve converges to the zero and the curve converges to the origin and is tangent to one of the origin and is tangent to one of the axesaxes



Type 2 systems: at Type 2 systems: at =0, the =0, the magnitude is infinity and the phase magnitude is infinity and the phase angle is -180angle is -180oo. At . At =∞ the magnitude =∞ the magnitude becomes zero and the curve is becomes zero and the curve is tangent to one of the axestangent to one of the axes


Polar plot of type 0, 1 and 2Polar plot of type 0, 1 and 2

= 0

∞

∞

∞

0

0

Type 0

Type 1

Type 2


Polar Plot of high freq. rangePolar Plot of high freq. range

n-m = 1

n-m = 1

n-m = 1


Nyquist in MatlabNyquist in Matlab

Use [re,im,w]=nyquist(sys)Use [re,im,w]=nyquist(sys) ‘‘sys’ may be filled with (num,den) or sys’ may be filled with (num,den) or

transfer function or (A,B,C,D) in state transfer function or (A,B,C,D) in state spacespace

In Nyquist, it is important to see the In Nyquist, it is important to see the direction of the curvedirection of the curve


Next…Next…

The Nyquist plot is already discussed, the subjects The Nyquist plot is already discussed, the subjects are to plot several general equation in Nyquist and are to plot several general equation in Nyquist and how to use Matlab to plot nyquist.how to use Matlab to plot nyquist.

The next class is Nyquist stability criterion and The next class is Nyquist stability criterion and phase-gain margin. Students have to prepare phase-gain margin. Students have to prepare themselves for this topic.themselves for this topic.

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

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