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6^TH Semester

Control System Engineering

Unit – 8

NYQUIST PLOT

By

Er.Gurupada Mishra

Nyquist Plot

• Nyquist Plots are a way of showing frequency responses of linear systems. 
• yquist plots display both amplitude and phase angle on a single plot, using frequency as a parameter in the plot. 
• Nyquist plots have properties that allow you to see whether a system is stable or unstable.

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Nyquist Stability Criterion

• The Nyquist stability criterion works on the principle of argument. It states that if there are P poles and Z zeros are enclosed by the ‘s’ plane closed path, then the corresponding G(s)H(s) plane must encircle the origin P−Z times. So, we can write the number of encirclements N as,

N = P – Z

• If the enclosed ‘s’ plane closed path contains only poles, then the direction of the encirclement in the G(s)H(s) plane will be opposite to the direction of the enclosed closed path in the ‘s’ plane.
• If the enclosed ‘s’ plane closed path contains only zeros, then the direction of the encirclement in the G(s)H(s) plane will be in the same direction as that of the enclosed closed path in the ‘s’ plane.

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POLAR PLOTS

To draw polar plot

ω increases, M decreases, φ increases negatively

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POLAR PLOTS

• The polar plot starts at a particular value when ω = 0 and ends at 0 when ω → ∞.
• The shape of polar plot depends on damping ration ( ).

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(

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POLAR PLOTS

• The intersection point on negative imaginary axis is nothing but the undamped natural frequency ω_n.
• For high damping ratio, the plot is closer to the real axis.
• The plot tends to become semicircle for overdamped system. Advantage It incorporates entire frequency range in the same plot. Disadvantage It does not clearly indicates the contribution of individual factors.
• Integral and derivative factor : → The polar plot G(jω) = 1/jω is the negative imaginary axis.

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BODE PLOTS

• The Bode plot for a linear time-invarient system with transfer function H(s) consists of a magnitude plot and a phase plot.
• The Bode magnitude plot is the graph of the function 

of frequency  . The  axis of the magnitude plot is logarithmic and the magnitude is given in decibels, i.e., a value for the magnitude |H| is plotted on the axis at 

• The Bode phase plot is the graph of the phase, commonly expressed in degrees, of the transfer function   as a function of .The phase is plotted on the same logarithmic  - axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.

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BODE PLOTS

The Bode plot or the Bode diagram consists of two plots

• Magnitude plot
• Phase plot

The magnitude of the open loop transfer function in dB is

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The phase angle of the open loop transfer function in degrees is

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BODE PLOTS

The following table shows the slope, magnitude and the phase angle values of the terms present in the open loop transfer function.

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Rules for Construction of Bode Plots

• Represent the open loop transfer function in the standard time constant form.
• Substitute,  in the above equation.
• Find the corner frequencies and arrange them in ascending order.
• Consider the starting frequency of the Bode plot as 1/10^th of the minimum corner frequency or 0.1 rad/sec whichever is smaller value and draw the Bode plot up to 10 times maximum corner frequency.
• Draw the magnitude plots for each term and combine these plots properly.
• Draw the phase plots for each term and combine these plots properly.

Note − The corner frequency is the frequency at which there is a change in the slope of the magnitude plot.

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Phase Cross over Frequency & Gain Margin

• The frequency at which the phase plot is having the phase of -180⁰ is known as phase cross over frequency. It is denoted by  . The unit of phase cross over frequency is rad/sec.
• Gain margin GM is equal to negative of the magnitude in dB at phase cross over frequency.

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Where,   is the magnitude at

phase cross over frequency.

The unit of gain margin (GM)

is dB.

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GAIN CROSS OVER FREQUENCY & PHASE MARGIN

• Gain cross over frequency is the frequency at which the magnitude of the G(s)H(s) becomes 1.

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• Phase margin is the phase that can be varied before the system becomes just stable (i.e., after varying the phase up to a certain threshold, the system becomes marginally stable and then further variation of phase leads to instability). Phase margin occurs at Gain Cross over frequency.

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STABILITY OF CONTROL SYSTEM

• If phase margin and Gain margin both are positive, the system is stable.
• If phase margin and Gain margin both are negative, the system is unstable.

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Performance Specification in Frequency

Transfer function of the second order closed loop control system

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where

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Magnitude of T(jω) is ,

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Phase of T(jω) is,

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Performance Specification in Frequency

• Resonant Frequency: It is the frequency at which the magnitude of the frequency response has peak value for the first time.

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• Resonant Peak: It is the maximum value of the magnitude of T(jω).

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Resonant peak in frequency response corresponds to the peak overshoot in the time domain transient response for certain values of damping ratio δ. So, the resonant peak and peak overshoot are correlated to each other.

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Performance Specification in Frequency

• Bandwidth: It is the range of frequencies over which, the magnitude of T(jω) drops to 70.7% from its zero frequency value.

At

At 3-dB frequency, the magnitude of T(jω) will be 70.7% of magnitude of T(jω) at ω = 0.

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Nyquist stability criterion applied to inverse polar plot

• If the Nyquist plot of , corresponding to the Nyquist contour in the s-plane encircles the point as many times as are the number of right half s-plane, poles of , the closed loop system is stable.
• In the special case where has no poles in the right half s-plane, the closed loop system is stable provided the net encirclement of point by the Nyquist plot of is zero.

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Effect of addition of poles and zeros to G(s)H(s) on the shape of Nyquist plot

• Addition of poles at s = 0

The addition of a pole at the origin to a loop transfer function rotates the Nyquist plot at zero and infinite frequencies by a further angle of . The addition of poles at to a loop transfer function will effect the stability of the closed loop system adversely.

• Addition of Finite non – zero poles

Addition of a non – zero pole to the loop transfer function shifts the phase of the Nyquist plot by at , i.e. it results in further rotation of the polar plot through an angle of as . The stability is adversely affected.

• Addition of a zero

Rotate the high frequency portion of the Nyquist plot by in the counter-clockwise direction without effecting the value at .

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Constant M and N circle

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Constant M and N circle

• Constant magnitude loci that are M-circles and constant phase angle loci that are N-circles are the fundamental components.
• The constant M and constant N circles in G (jω) plane can be used for the analysis and design of control systems.
• However the constant M and constant N circles in gain phase plane are prepared for system design and analysis as these plots supply information with fewer manipulations.
• Gain phase plane is the graph having gain in decibel along the ordinate (vertical axis) and phase angle along the abscissa (horizontal axis).

Constant M and N circle

• The M and N circles of G (jω) in the gain phase plane are transformed into M and N contours in rectangular co-ordinates.
• A point on the constant M loci in G (jω) plane is transferred to gain phase plane by drawing the vector directed from the origin of G (jω) plane to a particular point on M circle and then measuring the length in db and angle in degree.
• The critical point in G (jω), plane corresponds to the point of zero decibel and -180o in the gain phase plane. Plot of M and N circles in gain phase plane is known as Nichols chart /plot.

Thank You