# BSNL TTA MATCH OINT CONTROL SYSTEM (BODE PLOT AND NYQUIST CRITERION)

BSNL TTA MATCH OINT CONTROL
SYSTEM
(BODE
PLOT AND NYQUIST CRITERION)
BODE PLOT
These are also known as logarithmic plot (because we
draw these
plots on semi-log papers) and are used for determining the
relative
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 Bode phase plot.
Important Point Related to Bode plot-
1.Constant term K:
This factor has a slope of zero dB per
. There is no corner frequency corresponding to this constant term.
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 -20 dB per
. 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).
4. First order factor (1+jωT): This
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) +
{(1/ω2)}
× (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
phase
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).
2.
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.
4.
Calculate the gain factor and the
type or order of the system.
5.
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.
2.
Integral factor 1/(jω)n.
3.
First order factor 1/ (1+jωT).
4.
First order factor (1+jωT).
5.
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.
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.
3.
Calculate the phase margin.
1.
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.
3.
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.
5.
It also covers from low frequency to
high frequency range.
Various terms related to Bode Plot:
1.
Gain Margin: Greater will the gain margin greater will be 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.
2.
Phase Margin: Greater will the phase margin greater will be 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.
3.
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.
5.
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.
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
8.
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.
Stability Conditions of Bode Plots:
Stability
conditions are given below :
1.
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.
3.
For Unstable System : If any of them is negative. Or phase margin should
be less than the gain margin.

# NYQUIST PLOT

The stability analysis of a feedback control system
is based on
identifying the location of the roots of the
characteristic equation
on s-plane. The system is stable if the roots lie on
left hand side of
s-plane. Relative stability of a system can be
determined by using
frequency response methods like Nyquist plot
and Bode plot.
Nyquist criterion is used to identify the presence
of roots of a
characteristic equation in a specified region of
s-plane.

## Nyquist path or Nyquist contour:

The Nyquist contour is a closed
contour in the s-plane which
completely encloses the entire right hand half of
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.

## Nyquist Encirclement:

A point is said to be encircled by a contour if it
is found inside the
contour.

## Nyquist Mapping:

The process by which a point in s-plane transformed
into a point in
F(s) plane is called mapping and F(s) is called
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 the
total no. of encirclement about the origin, P is the
total no. of poles
and Z is the total no. of zeroes.
Case 1:-
N = 0 (no encirclement), so Z = P = 0 & Z = P If N = 0, P
must be zero therefore system is stable.
Case 2:- N > 0
(clockwise encirclement), so P = 0 , Z ≠0 & Z > P
For both cases system is unstable. Case 3 :- N < 0
(counterclockwise encirclement), so Z = 0, P ≠0
& P > Z System
is stable.