BSNL TTA MATCH OINT CONTROL

SYSTEM

SYSTEM

(BODE

PLOT AND NYQUIST CRITERION)

PLOT AND NYQUIST CRITERION)

BODE PLOT

These are also known as logarithmic plot (because we

draw these

draw these

plots on semi-log papers) and are used for

**determining the****relative**

stabilities of the given system.

stabilities of the given system.

In

order to determine the stability of the system using bode plot we draw two curves,

one is for magnitude called magnitude curve another for phase called

order to determine the stability of the system using bode plot we draw two curves,

one is for magnitude called magnitude curve another for phase called

**Bode****phase plot**.
Important Point Related to Bode plot-

1.Constant term K:

This factor has a slope of

The phase angle associated with this constant term is also zero.

This factor has a slope of

**zero dB per**

decade. There is no corner frequency corresponding to this constant term.decade

The phase angle associated with this constant term is also zero.

2. Integral factor 1/(jω)n: This factor has a slope of

**-20 × n (where n is any****integer)dB per decade.**There is no corner frequency corresponding to this integral

factor. The phase angle associated with this integral factor is -90 × n here n

is also an integer.

3. First order factor 1/ (1+jωT):

This factor has a slope of

second. The phase angle associated with this first factor is -tan- 1(ωT).

This factor has a slope of

**-20 dB per**

decade. The corner frequency corresponding to this factor is 1/T radian perdecade

second. The phase angle associated with this first factor is -tan- 1(ωT).

4. First order factor (1+jωT): This

factor has a slope of

The corner frequency corresponding to this factor is 1/T radian per second. The

phase angle associated with this first factor is tan- 1(ωT) .

factor has a slope of

**20 dB per decade**.The corner frequency corresponding to this factor is 1/T radian per second. The

phase angle associated with this first factor is tan- 1(ωT) .

5.Second

order or quadratic factor : [{1/(1+(2ζ/ω)} × (jω) +

order or quadratic factor : [{1/(1+(2ζ/ω)} × (jω) +

{(1/ω2)}

× (jω)2)]:

× (jω)2)]:

This factor has a slope of

**-40 dB per decade**. The corner
frequency

corresponding to this factor is ωn radian per second. The

corresponding to this factor is ωn radian per second. The

phase

angle associated with this first factor is – tan-1{ (2ζω / ωn)

/

angle associated with this first factor is – tan-1{ (2ζω / ωn)

/

(1-(ω / ωn)2)}

.

.

Procedure of making a bode plot:

1.

Substitute the s = jω in the open

loop transfer function G(s) × H(s).

Substitute the s = jω in the open

loop transfer function G(s) × H(s).

2.

Find the corresponding corner

frequencies and tabulate them.

Find the corresponding corner

frequencies and tabulate them.

3.

Now we are required one semi-log

graph chooses a frequency range such that the plot should start with the

frequency which is lower than the lowest corner frequency. Mark angular

frequencies on the x-axis, mark slopes on the left hand side of the y-axis by

marking a zero slope in the middle and on the right hand side mark phase angle

by taking -180 degrees in the middle.

Now we are required one semi-log

graph chooses a frequency range such that the plot should start with the

frequency which is lower than the lowest corner frequency. Mark angular

frequencies on the x-axis, mark slopes on the left hand side of the y-axis by

marking a zero slope in the middle and on the right hand side mark phase angle

by taking -180 degrees in the middle.

4.

Calculate the gain factor and the

type or order of the system.

Calculate the gain factor and the

type or order of the system.

5.

Now calculate slope corresponding to

each factor.

Now calculate slope corresponding to

each factor.

For drawing the

**Magnitude curve :**- Mark the corner frequency on the semi log graph paper.
- Tabulate these factors moving from top to bottom in the

given sequence.

1.

Constant term K.

Constant term K.

2.

Integral factor 1/(jω)n.

Integral factor 1/(jω)n.

3.

First order factor 1/ (1+jωT).

First order factor 1/ (1+jωT).

4.

First order factor (1+jωT).

First order factor (1+jωT).

5.

Second order or quadratic factor :

[{1/(1+(2ζ/ω)} × (jω) + {(1/ω2)} × (jω)2)]

Second order or quadratic factor :

[{1/(1+(2ζ/ω)} × (jω) + {(1/ω2)} × (jω)2)]

- Now sketch the line with the help of corresponding

slope of the given factor. Change the slope at every corner frequency by

adding the slope of the next factor. You will get magnitude plot. - Calculate the gain margin. For

drawing the**Bode phase plot**:

1.

Calculate the phase function adding

all the phases of factors.

Calculate the phase function adding

all the phases of factors.

2.

Substitute various values to above

function in order to find out the phase at different points and plot a curve.

You will get a phase curve.

Substitute various values to above

function in order to find out the phase at different points and plot a curve.

You will get a phase curve.

3.

Calculate the phase margin.

Calculate the phase margin.

**Advantages of Bode Plot:**

1.

It is based on the asymptotic

approximation, which provides a simple method to plot the logarithmic magnitude

curve.

It is based on the asymptotic

approximation, which provides a simple method to plot the logarithmic magnitude

curve.

2.

The multiplication of various

magnitude appears in the transfer function can be treated as an addition, while

division can be treated as subtraction as we are using a logarithmic scale.

The multiplication of various

magnitude appears in the transfer function can be treated as an addition, while

division can be treated as subtraction as we are using a logarithmic scale.

3.

With the help of this plot only we

can directly comment on the stability of the system without doing any

calculations.

With the help of this plot only we

can directly comment on the stability of the system without doing any

calculations.

**4.**

**Bode plots**

**provides**

relative stability in terms of gain

margin and phase margin.

relative stability in terms of gain

margin and phase margin.

5.

It also covers from low frequency to

high frequency range.

It also covers from low frequency to

high frequency range.

Various terms related to Bode Plot:

1.

Gain Margin: Greater will the

stability of the system. It refers to the amount of gain, which can be

increased or decreased without making the system unstable. It is usually

expressed in dB.

Gain Margin: Greater will the

**gain margin**greater will be thestability of the system. It refers to the amount of gain, which can be

increased or decreased without making the system unstable. It is usually

expressed in dB.

2.

Phase Margin: Greater will the

stability of the system. It refers to the phase which can be increased or

decreased without making the system unstable. It is usually expressed in phase.

Phase Margin: Greater will the

**phase margin**greater will be thestability of the system. It refers to the phase which can be increased or

decreased without making the system unstable. It is usually expressed in phase.

3.

Gain Cross Over

Frequency: It refers to the frequency at which

magnitude curve cuts the zero dB axis in the bode plot.

Gain Cross Over

Frequency: It refers to the frequency at which

magnitude curve cuts the zero dB axis in the bode plot.

4.

Phase Cross Over

Frequency: It refers to the frequency at which

phase curve cuts the negative times the 180 degree axis in this plot.

Phase Cross Over

Frequency: It refers to the frequency at which

phase curve cuts the negative times the 180 degree axis in this plot.

5.

Corner Frequency: The frequency at which the two asymptotes cuts or meet each

other is known as break frequency or corner frequency.

Corner Frequency: The frequency at which the two asymptotes cuts or meet each

other is known as break frequency or corner frequency.

6.

Resonant Frequency: The value of frequency at which the modulus of G (jω) has a

peak value is known as resonant frequency.

Resonant Frequency: The value of frequency at which the modulus of G (jω) has a

peak value is known as resonant frequency.

7.

Factors: Every loop transfer function (i.e. G(s) × H(s)) product of

various factors like constant term K, Integral factors (jω), first order

factors ( 1 + jωT)(± n) where n is an integer, second order or

quadratic factors.

Factors: Every loop transfer function (i.e. G(s) × H(s)) product of

various factors like constant term K, Integral factors (jω), first order

factors ( 1 + jωT)(± n) where n is an integer, second order or

quadratic factors.

8.

Slope: There is a slope corresponding to each factor and slope for

each factor is expressed in the dB per decade.

Slope: There is a slope corresponding to each factor and slope for

each factor is expressed in the dB per decade.

9.

Angle: There is an angle corresponding to each factor and angle

for each factor is expressed in the degrees.

Angle: There is an angle corresponding to each factor and angle

for each factor is expressed in the degrees.

**Stability Conditions of Bode Plots:**

Stability

conditions are given below :

conditions are given below :

1.

For Stable System: Both the margins should be positive. Or phase margin should

be greater than the gain margin.

For Stable System: Both the margins should be positive. Or phase margin should

be greater than the gain margin.

2.

For Marginal Stable

System : Both the margins should be zero. Or

phase margin should be equal to the gain margin.

For Marginal Stable

System : Both the margins should be zero. Or

phase margin should be equal to the gain margin.

3.

For Unstable System : If any of them is negative. Or

be less than the gain margin.

For Unstable System : If any of them is negative. Or

**phase margin**shouldbe less than the gain margin.

# NYQUIST PLOT

The stability analysis of a feedback control system

is based on

is based on

identifying the location of the roots of the

characteristic equation

characteristic equation

on s-plane. The system is stable if the roots lie on

left hand side of

left hand side of

s-plane. Relative stability of a system can be

determined by using

determined by using

frequency response methods like

and Bode plot.

**Nyquist plot**and Bode plot.

Nyquist criterion is used to identify the presence

of roots of a

of roots of a

characteristic equation in a specified region of

s-plane.

s-plane.

## Nyquist path or Nyquist contour:

The

contour in the s-plane which

**Nyquist contour**is a closedcontour in the s-plane which

completely encloses the entire right hand half of

s-plane. In order

s-plane. In order

to

enclose the complete RHS of s-plane a large semicircle path is drawn with

diameter along jω axis and centre at origin. The radius of the semicircle is

treated as infinity.

enclose the complete RHS of s-plane a large semicircle path is drawn with

diameter along jω axis and centre at origin. The radius of the semicircle is

treated as infinity.

##
Nyquist

Encirclement:

A point is said to be encircled by a contour if it

is found inside the

is found inside the

contour.

##
Nyquist

Mapping:

The process by which a point in s-plane transformed

into a point in

into a point in

F(s) plane is called mapping and F(s) is called

mapping function.

mapping function.

###
Steps of drawing the Nyquist

path:

- Step 1 – Check for the poles of G(s) H(s)

of jω axis including that at origin. - Step 2 – Select the proper Nyquist contour

– a) Include the entire right half of s-plane by drawing a semicircle of

radius R with R tends to infinity. - Step 3 – Identify the various segments on

the contour with reference to**Nyquist path** - Step 4 – Perform the mapping segment by

segment substituting the equation for respective segment in the mapping

function. Basically we have to sketch the polar plots of the respective

segment. - Step 5 – Mapping of the segments are

usually mirror images of mapping of respective path of +ve imaginary axis. - Step 6 – The semicircular path which covers

the right half of s plane generally maps into a point in G(s) H(s) plane. - Step 7- Interconnect all the mapping of

different segments to yield the required**Nyquist diagram**. - Step 8 – Note the number of clockwise

encirclement about (-1, 0) and decide stability by N = Z – P -
**Nyquist stability criterion**:

says that N = Z – P. Where, N is thetotal no. of encirclement about the origin, P is the

total no. of polesand Z is the total no. of zeroes.Case 1:-

N = 0 (no encirclement), so Z = P = 0 & Z = P If N = 0, Pmust be zero therefore system is stable.Case 2:- N > 0

(clockwise encirclement), so P = 0 , Z ≠0 & Z > PFor both cases system is unstable. Case 3 :- N < 0(counterclockwise encirclement), so Z = 0, P ≠0

& P > Z Systemis stable.NOTE-**±20Ndb/decade=±20×0.30=±6Ndb/octave****YOU CAN DOWNLOAD IT FROM PAGE STUDY MATERIAL**

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