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I think we can now resume our discussion on the Nyquist stability criterion. The use of
the criterion for study of absolute stability has already been taken. Now let us take how
to use the criterion for the study of relative stability. You already know what relative
stability is. Let me take you to the time domain; let us say this is a time domain plot
and the closed-loop this is sigma j omega s plane I am taking these are the two closed-loop
poles the dominant poles I have taken.
Now if you take sigma and j omega axis here and the two poles if they are taken close
to the j omega axis you know that the transient in this particular system will die faster
compared to this particular case and therefore you can say that the relative stability of
the system is guided by the distance of the real part of these poles with respect to the
imaginary axis. You already know that this real part guides the envelop of the oscillating
response. So we know that the closer the dominant poles are to the j omega axis poorer is the
relative stability of the system.
Let us interpret this in frequency domain. In frequency domain, if I consider this system
and make a Nyquist plot of it this is now the GH plane that is G(s) H(s) plane where
GH is the open-loop transfer function of the system. So this is real part, this is imaginary
part, this is minus 1. When I say the Nyquist plot primarily you see it is the polar plot
because its mirror image about the real axis and then closing the path will give you the
complete Nyquist plot.
In this particular case please see I expect that the plot will be of this nature. For
example, just I am giving you an example, the plot will be of this nature while in this
particular case I expect this is real, this is imaginary, this is your minus 1 point the
plot will be of this nature. What is the different between the two; how do I evaluate which system
is better? You please see that the enclosure of the minus 1 point by the Nyquist plot is
the index of absolute stability. Now you can, I think, initiatively feel that closer this
plot is to the minus 1 point more is the risk of system going towards instability as is
the case over here that is here the index was the distance with respect to imaginary
axis.
Now here I can say that the polar plot portion, you see that if I give you a complete Nyquist
plot you will realize that primarily we are interested in the polar plot; this is real
part, this is imaginary part, this is minus 1 plus j0 a typical system, this is omega
is equal to 0 plus, this is omega is equal to plus infinity, I have omega is equal to
minus infinity and going to omega is equal to 0 minus and this is getting closed.
So you can see that the primary contribution to this particular plot is given by the polar
plot, it is the frequency response of the system which you have got from omega is equal
to 0 to omega is equal to infinity. So you can very well see, in this particular case,
if the system goes this way this is more closer to the minus 1 point and therefore there is
more risk of system going to sustained oscillations that is the system is more prone to instability.
If you increase the gain so that it goes this way and naturally the system has become unstable
because now minus 1 plus j0 point is enclosed provided I am assuming that the P is equal
to 0, please see. Stability is definitely with respect
to the number of open-loop poles in the right-half plane. So if I am analyzing the stability
with respect to the Nyquist plot given to you I assume that P is equal to 0 and therefore
this system is more stable, this system is more closer to instability or sustained oscillations
and this system becomes unstable. So I can say that the relative stability is given by
relative distance of the plot the polar plot or the Nyquist plot with respect to the point
minus 1 plus j0.
Now how to measure relative distance? Let us say that this is minus 1 point, if the
plot intersects at this particular point, well there is a big distance as far as the
real axis is concerned. So it means I consider this to be the intersection point and let
us say that this is the magnitude a so it appears lower the magnitude a more is the
stability achieved because the intersection is closer to this and hence we are far away
from the minus 1 plus j0 point. See this particular situation. So naturally we are far away from
minus 1 plus j0 point if this intersection is smaller this magnitude is small.
But you see the situation; after all this is a vector, it is not only a magnitude it
depends up on the phase angle as well. Let me take this intersection but let me assume
that the plot is like this, after all this is also a possibility, this is also a polar
plot of a typical system. So in this particular case though the intersection is same though
we are far away from the minus 1 plus j0 point as far as the intersection point is concerned
but still you find that the green curve is more prone to instability because it is closure
to minus 1 plus j0 point so it appears that the distance the relative distance of the
polar plot with respect to minus 1 plus j0 point cannot be given by just the intersection
with the negative real axis the phase angle is also important and that is why two indices
are used.
I am giving these two indices; two indices are used to measure the distance and the indices
are this is minus 1 point this is minus 1 point 1 is this point of intersection let
me call it (a). (a) is the magnitude, this is one index and the other index, you see
the phase angle, what is done is you see that you just draw a unit circle with this origin
as the centre that is you locate a point here on this particular curve where the magnitude
is 1, this particular angle is also a measure of closeness of the plot to minus 1 plus j0
point.
I again repeat; the magnitude point on the plot which is equal to unity is located which
is equivalent to saying that a unit circle passing through minus 1 point wherever it
intersects your polar plot is the critical point you want to consider. So this particular
point is taken and this particular point you draw a line over here this angle is the angle
which gives you a measure of the distance with respect to minus 1 plus j0 point.
Yes please,.........but if it not intersect at all, it may not intersect at all, okay
those points also we will see. I will keep your point in mind and I will answer this.
So if I consider this very plot which I have given you where there is an intersection over
here in that particular case you find that this distance and this angle the two you see
you can claim that, well, even these two may not be sufficient so naturally the complete
plot is a more better description of its distance with respect to minus 1 plus j0 point. But
if I take the complete plot I cannot set the complete plot into suitably I to my design
algorithm. So after all I want a simple design algorithm therefore the complete plot has
been quantified into two indices: one is the intersection with this axis and the other
is this particular angle where the magnitude is 1.
Now you just see, if I increase the magnitude, please see, angles will remain the same, if
I increase the magnitude what type of response do you get, well, this is let us say at a
value of K higher than.... this is K 2 which is greater than K 1 all the angles will be
remaining the same but only the magnitudes will be changing, keep all the polar vectors
only change the magnitudes, and if you further increase your magnitude the plot will be like
this that is it is the same type of plot the only thing is that its magnitudes change and
therefore the plot goes this way. So it means there will be an increase of gain which will
slowly take it through the minus 1 plus j0 point.
Can you tell me by what factor can I increase the gain so that the plot passes through minus
1 plus j0 point. I repeat my question; by what factor can I increase the gain so that
the plot passes through minus 1 plus j0 point? it is 1 by a obviously, please see, if (a)
is the intersection here so you want this magnitude at this particular frequency this
is one vector minus 180 degrees. So this will pass through minus 1 point it means the K
is to be elongated by this much factor which equal to 1by a. So it means if your original
polar vectors all of them are multiplied by 1 by a in that particular case the plot will
pass through minus 1 plus j0 point and this 1 by a factor is referred to as gain margin
GM it is a very very important term for us. So gain margin is 1 by a it means this much
margin is available to you before you drive the system to marginal stability or towards
instability. So this is the gain margin.
Similarly, please see, let me take this sketch again, this was a typical plot I am taking,
all the specific point I will answer later. you see this is the point, this is the magnitude
a and let us say that this plot corresponds to G(s) H(s) equal to (1 plus s T 1) s into
1 plus s T 2 some function I have taken: 1 plus s T 1 okay you can take even some K 1
over s into 1 plus s T 2 is the transfer function for which the polar plot is given to you;
omega is equal to 0 to omega is equal to infinity. You will note that I have taken P is equal
to 0 that is the right-half plane poles is equal to 0.
Now you can see if I increase the gain by a factor of 1 by a in that particular case
the system will pass through minus 1 plus j0 point and hence will become marginally
stable at the most marginally stable and therefore corresponding to this particular situation
I can say that the gain margin of the system is 1 by 8. Consider the phase margin; you
see this is the point, take this angle, you see how much angle you can add, how much phase
lag, you see phase lag is the term I will use which creates in stability. You have seen
dead time introduces phase lag which is a problem which creates instability. So you
see that the question is how much phase lag can be added to the system so that the system
this particular point passes through minus 1 plus j0 point
in that particular case that I will say is the phase margin and you can see that this
much is the additional phase lag which you can add to this system. So the phase margin
is positive.
Suppose this phi it means plus phi is the phase margin because by dead time or because
of any other factor in the loop if an additional phase lag of minus phi appears into the system
the system still remains stable that is it just goes to the verge of instability. You
will please note that this phase margin is positive because this much of phase margin
still you have, if this much of phase lag is added to the system in that particular
case the system goes to the point of instability and hence these are the two measures the gain
margin and the phase margin which represent the measure of relative stability using the
Nyquist stability criterion, very important measures.
Let me answer his point. He said that, well, the plot may not intersect, I can even give
you the example, you know that example where the plot will not intersect, this is the example,
yes, what is the polar plot of this system? The polar plot of this particular system as
you know is this; this is your minus 1 plus j0 point. Now you increase the value of K,
any amount you increase the value of K it has to become asymptotic to this particular
line and since for every value of K the plot is asymptotic to this particular line in that
particular case you can say that the plot will never intersect. Hence you can say the
gain margin in this particular case is infinity.
But you have not to be very happy that the gain margin is infinity, it simply says that
for such systems where the intersection is not there phase margin is the prime index
of relative stability of a system because it tells you how close it goes to minus 1
plus j0 point. So your gain margin is infinity in this particular case and the phase margin
you can see will depend upon the value of K. So this much is the phase margin here the
phase margin is reducing, phase margin is reducing, phase margin has further reduced
as you keep on increasing the value of K while in every case your gain margin is equal to
infinity.
.......Sir,..... also not intersect this may not intersect, yes, come on, cite some example,
((constant is could be included or)) this is in hypothetical case, come up with the
case, this question was there, yes, I know that this is a physical system, this is just
a hypothetical case and if you come up with the case I think we will evaluate with that
particular case otherwise we do not come across such situations where this does not intersect
because omega is equal to 0 to infinity it will pass through this.
Thus, now let me say that this discussion is complete. .......... Excuse me sir, yes
please..... Sir, if the numerator and denominator degrees are equal then that case at s equal
to infinity it will not tend to origin it tends to some constant in that case the circle
might not intersect the plot)) okay let us take an example.
I take an example G(s) H(s) equal to....... Well, you are referring to a type-0 system
that is all, I can tell you this thing that you are refereeing to a type-0 system so let
me take 1 over (1 plus s T 1) (1 plus s T 2) Sir, talking about the case where numerator
and denominator are equal.... 1 pulse s T 1 divided by 1 plus s T 2 come on let us quickly
get the rough plot of this; 1 plus j omega T 1 over 1 plus j omega T 2, help me please,
what is the value at omega is equal to 0 this is your starting point. When omega tends to
infinity when omega tends to infinity what is your point in that particular case it is
j omega T 1 over j omega T 2. So in this particular case there is certain magnitude in that magnitude
you can see this T 1 by T 2 angle will be 0 and magnitude you can take as going to certain
finite value.
Now it depends about the intersection with this axis......... come on, think of the intersection
with this axis or should it go this way or it is the simple plot? It depends, yes, could
you help me please; I think this may be simple plot. In this particular case if I take 1
plus j omega T 1......... it does not intersect with the imaginary axis does not intersect
with the imaginary axis in this case this is a simple plot and if I take the phase margin
in this particular case the phase margin definition, saying that phase margin is equal to any value......
well, I mean, the only thing in this particular case is this that the gain margin and phase
margin the system is stable for all values of T 1 and T 2 the only thing it can be said
that.... yes, I think I will study this point again, it has come to me, sir we have an example,
yes come on......... s plus half divided by s plus 1 into s minus 1 s plus 1........ no,
this case I will not take, it is a case of an open-loop unstable system I am going to
comment on this; open-loop unstable system will not fit into the definition of the gain
margin and phase margin I have given. I will explain this point very shortly.
This example seems to be quite alright that here we have a type-0 system and in the type-0
system we have a numerator 0 and in this particular case if I vary T 1 and T 2 so the system is
stable for all values of T 1 and T 2. The relative stability we are studying with respect
to certain parameters, the only reply which immediately comes to me, you see, though the
quantitative reply will come later the immediate reply in this particular case which comes
to my mind is this that since we have taken this example and T 1 and T 2 are the two parameters;
we are studying the relative stability of the system with respect to the parameter of
the model whether it is gain or whether it is any other parameter and we say that the
parameters of the model can be changed because this much of gain margin or this much of phase
margin is available.
So in this particular case if you study the stability of the system it will turn out to
be stable for all values of T 1 and T 2. Therefore it appears the term gain margin and phase
margin for such a system lose meaning. This is the tentative reply I am giving you. But
mathematically or quantitatively I think I will be able to answer you next time. But
if you have any point please come up with that. This is the thing. From the plot which
we have got it appears that for all values of T 1 and T 2 the system is stable so you
are studying the relative stability of the system with respect to the parameters of the
system that is how the parameters of the system can drive the system to the verge of instability.
Therefore, in this particular case since the parameter of the system can never drive this
system to instability so may be the indices the gain margin and the phase margin lose
their meaning or you may say that, well they are infinite but there it might just turn
out be a mathematical definition.
For example, gain margin is equal to infinity is it not a mathematical definition only because
that type of system a type-0 system we had taken type-1 system we had taken gain margin
is equal to infinity does not really mean that we have a very good stable system. Naturally
the phase margin was important there. Similarly, in this particular case it appears that the
system is stable for all parameter values of T 1 and T 2 and therefore the definition
is to be reviewed. I think we can keep this point pending unless there is an additional
point from your side. Hopefully the answer will be right which I am giving.
Now let me answer his point. His point is this; he has given this example: 1 over s
plus half s plus 1 into s minus 1 please see that when you have an open-loop pole. Open-loop
pole is unstable; in this, the open-loop system is unstable in that particular case your gain
margin and phase margin definitions change because in this case now you see for stability
of the system itself your system has to encircle the point minus 1 plus j0. One encirclement
you will require because there is P is equal to 1 so the gain margin and phase margin definitions
which I have given you earlier they do not carry over straightaway to a system which
is open-loop unstable because in this particular case this plot if I make..... let us assume
though in this particular case it will not be a plot for this let me put some s also
because then it may take this shape; so let me assume that it is this shape in that particular
case the system will be stable only if there is one encirclement in the counter clockwise
direction and that encirclement corresponds to the open-loop pole in the right-half plane.
Not that the definitions of stability margins, not that the definition of gain margin and
phase margin cannot be extended to the situations where the open-loop system is unstable but
normally we do not do it if the open-loop system is unstable we go to the complete Nyquist
plot, we go the root locus analysis, we go to the other methods of relative stability
analysis, we normally do not use the stability margins as an index of relative stability.
I am explaining this point not that it is not possible to define but it loses its advantage
you see.
For every case if there are two poles for that case you will have to take special care
that is for the two poles there have to be two encirclements of the minus 1 plus j0 point
for the system to be stable and hence let us conclude that the stability margins the
definition of gain margin and the phase margin which I have given these definitions are applicable
to open-loop stable systems. We limit our discussions to this and this limitation is
not a big limitation because most of the industrial systems we work with are open-loop stable
systems. So if you come across an open-loop unstable system take special care of that
system that is the only point.
Now let us just finally take up the definition of the gain margin and phase margin, the special
cases also we have taken where there are no intersections we have taken. Now let me take
a general case; let us say a typical plot let me take this as minus 1 plus j0. Couple
of more definitions could also come. You will see that this is the point at which the angle
is minus 180 degrees, this is the point at which the angle is minus 180 degrees so let
me call this as omega PC omega phase crossover. This is omega phase crossover that is it is
the phase crossover frequency, it is the frequency at which the angle is minus 180 degrees. these
terms will be very often used and hence should be clear phase crossover frequency.
Similarly, let me take the point at which the magnitude is 1 unit magnitude and let
me call this frequency as omega gc gain crossover frequency. So I have to determine, this is
for you to determine the two frequencies: the gain crossover and the phase crossover
and this you can determine this particular plot the Nyquist plot or the polar plot is
available to you either from the experimental data or from the data derived from the open-loop
transfer function of the system. So these two definitions please.
Now the other definition; take the magnitude of G(j omega) G(j omega) H(j omega) magnitude
at omega equal to omega phase crossover. Analytically also you can obtain the gain margin phase
margin please see. At this particular frequency take the magnitude and say that this magnitude
is (a) in that particular case your gain margin is equal to 1 by a.
Thus, this is the gain margin please, the original gain of the system for which this
plot has been made, and this point may please be noted. It is not the gain of the system;
it becomes the gain only when the original plot has been made for a unity gain otherwise
it represents the factor by which the original gain of the system can be changed.
So lastly let me explain the phase margin and here I will draw a line; I will measure
this angle and this angle I will call the phase margin. This is the margin by which
the phase can be increased and now you can see that, tell me know, I think if the definition
is clear and if it is clear we are working with open-loop stable systems, tell me, for
a stable system the gain margin will be positive or negative for a stable system, the gain
margin will be greater than 1 or less than 1?
Please see that this point may please be noted very carefully; the gain margin will always
be greater than 1 for a stable system and it will be less than 1 for an unstable system
surely. You see that it crosses this. Now surely it is unstable because we are working
with open-loop stable cases. So it means if it crosses this and still you
apply your definition of the gain margin so if it is less than 1 so it means the gain
has to be, I mean qualitatively what is the meaning of the gain margin is less than 1
so it simply means that you have already increased the gain to an extent that the system has
become unstable; now you decrease the gain by this factor if you want to bring the system
back to the stability region. This point may please be noted. Your system has already been
driven to instability region because of higher value of gain you reduce the value of gain
so that the system comes back to the stability region. So the gain margin is greater than
1 for stability less than 1 for unstable systems.
Similarly, you can see that the phase margin.......... for this particular case if you see the unit
point the unit circle is here it crosses this here as far as the this plot is concerned
and this will become your phase margin. So it means if I go this way I measure this angle
in the clockwise direction and the angle is negative, if I go this way I measure this
angle in the counter clockwise direction and the angle is positive.
So you can now see that keeping this convention that the angle will always be measured with
respect to minus 180 degree axis and it will be positive if I go in the counter clockwise
direction it will be negative if I go in the clockwise direction then I can say that if
the phase margin is positive it is an indication of a stable system and if phase margin is
negative it is an indication of an unstable system. So these are the qualitative indices
and you will find that the phase margin and the gain margin they become two very important
performance specifications of a system for design in frequency domain. That is why it
is said that the design in frequency domain is centered around the gain margin and the
phase margin.
Now, under the Nyquist stability criterion the last point which is left is the use of
Bode plots for gain margin phase margin analysis; for stability analysis use of Bode plots.
Now, in this particular case please note that the Nyquist plot requires plotting of G(j
omega) H(j omega) which as you know is magnitude G(j omega) H(j omega) angle G H(j omega);
it means you require a table omega the magnitude and the angle for frequencies from 0 to infinity
this will give you the polar plot. So it means lot of calculation is involved. And if this
much of calculation is being done in that particular case it is all the time better
to go to the computer and immediately get the polar plot of the system and hence the
Nyquist plot of the system.
Fortunately we have a method; Bode suggested this method by which this particular plot
the frequency response data can be displayed on plot without hardly any calculation, without
any calculation it will become a straight line asymptotic plot and you will not require
this calculation omega, magnitude and phase angle and all your conclusions about the system
stability, the gain margin and phase margin will be easily interpreted on the Bode plots.
So the contribution of Bode is an extremely important contribution because the frequency
response design the frequency domain design is done on the Bode plot and not on the Nyquist
plot. The two ideas are getting mixed you see. The basic idea is given by Nyquist the
Nyquist stability criterion but the use of the idea has given by Bode in terms of the
Bode plot.
Therefore, now onwards you will hardly come across Nyquist plot in our discussion because
a tool which is easier than the polar plot will become in our hand as soon as I give
you the constructional methods and analysis methods on the Bode plots. So now let me introduce
the Bode plot to you and I introduce it with respect to this transfer function G(j omega)
a typical transfer function (1 plus j omega T 1) j omega (1 plus j omega T 2) and other
factors here as well as here. Take up these factors; simple factors I have taken but all
other factors can easily be taken.
Therefore, now you see that I have taken this to be a type-1 system, a square will come
if I consider a type-2 system and so on. In general it will be an nth power giving you
a type-M system. So magnitude and phase is to be calculated. But you see what will happen
if I take the log of this particular equation. If I take the log on both sides you see that
these multiplicative terms they become additive. So in that particular case I can say log of
base 10 G(j omega) H(j omega) is equal to log base 10 K plus log base 10 (1 plus j omega
T 1) minus log base 10 j omega minus log base 10 (1 plus j omega T 2) and so on. This is
the basic idea.
Now we can say that it is known to us what is their beginning but he said that, well,
if we take log, not only this but one more contribution that he will take appropriate
scales on the magnitude on the vertical and the horizontal axis so that all these individual
contributions become straight lines please see. not only the logs, additional contribution
from Bode is this that the plot of all these terms are more or less straight lines and
therefore addition of all these straight lines becomes easier and hence the overall plots
is easily available to you. So this is the major thing that the multiplicative terms
become additive terms if you take the log operation on the transfer function.
What I will do now, I will take the magnitude and phase separately; the magnitude of this
as you see is K 1 plus j omega T 1 magnitude j omega magnitude 1 plus j omega T 2 magnitude.
I hope you will not mind if I do n't write additional terms; when I take the general
case I will be able to write the additional terms also quickly without any difficulty.
This is magnitude, how about the angle please? Angle G(j omega) H(j omega) is equal to........
help me please, as far as the angle contribution of K is concerned it is 0 so it is minus it
is plus tangent inverse omega T 1 minus 90 degrees minus tangent inverse omega T 2 this
becomes the angle. So take the log of the magnitude. Since the angle is already a linear
term you can see that the angle is already a summation of various terms. Let me take
the log of the magnitude: log 10 I can take, no, forget about it because it will be 10
all through in my discussion so: G(j omega) H(j omega) magnitude is equal to log of K
plus log of (1 plus j omega T 1) magnitude minus log of j omega magnitude minus log of
1 plus j omega T 2 magnitude.
Now you see, I do not find, at least I do not find any justification in using the units
in control systems, for this magnitude the unit is decibel, I will define the decibel.
Decibel is used, it is the unit abbreviated as db and if you have a magnitude M unit-less
magnitude M then 20 log to the base M 10 of M is equal to the magnitude in decibels and
this is what we take, the log of magnitude has been taken, the unit we take is decibels;
now you can say that why we multiply by 20, the only reason probably comes to my mind
is that, in communication engineering right from the beginning from bel we went to decibel
the decibel unit depending upon the power of communication was a suitable unit used
so we are continuing with it and there is no reason to revert back because reverting
back also does not give you any advantage that is probably the only reason otherwise
you could say that log of 10 log base 10 of magnitude M could directly be the magnitude;
this 20 factor does not give you any advantage quantitatively other than this that you are
using a well-known unit this is my view point and as it has been said in the literature.
One more point please; one question which comes to my mind is that why do we take log
to the base 10 why not a natural log. You see that even the natural log will break it
up. Again the argument probably is the same. The natural log also will break it up and
will convert the multiplicative terms into the additive terms and you can develop the
entire thing in terms of the natural log but since the unit of this is decibels and Bode
has developed all along in terms of these units even reverting to the natural log does
not give you any specific advantage. So I say that the base 10 and the multiplication
by 20 taking it to decibel is just because of historical reasons and there is no other
qualitative reason which can be attributed to it.
Therefore, if we agree to this that we will also continue with this in that particular
case please see the angle becomes this and the magnitude 10 I can drop altogether now
throughout my discussion because l o g will mean that the base is 10. So I can now multiply
it by 20, this unit multiplied by 20, this unit multiplied by 20, 20, and 20 and so on
and this is the decibels.
So it means what is the Bode's contribution, what is the Bode plot? The Bode plot now will
consist of two plots instead of one as in the case of polar plot. you see, first let
me not take the Bode plot let me take the possibility of the plots the possibility of
the plots could be you take omega the frequency on this side and decibels on this side you
call it a magnitude plot, you take omega the frequency and the phase angle phi on this
side you call it a phase plot.
Now you still if you plot it this way you do not get any big advantage, you see. Though
the multiplication has been converted into addition in addition to this you do not get
any advantage. However, as you will see as I will show to you if instead of omega if
you take log omega as the horizontal axis, if you take log omega as the horizontal axis
then all the individual terms I have given you in the previous slide they are actually
straight lines with respect to log omega and not with respect to omega. So it means their
plots are going to be simple, making the plots are going to be simple so log omega is on
this axis and decibels.
Now what will be the scale on this side? The scale on this is going to be linear and
not log because the log operation has already been taken and the magnitude now decibels
can be taken with respect to linear scale. So this scale is linear on this axis, this
scale is log omega scale, here phase is linear on this axis and here omega is the log omega
scale. In that particular case you can see that the asymptotic plots or the approximate
plot in both the cases phase as well as this becomes the straight line asymptotes.
You could see that, well, this point comes to my mind that why do you take the log operation
first. You take the terms and you can use a scale which is a log log scale or before
I answer this let me take this semi log paper, I have checked, it cannot be projected it
is not visible is it, that is fine; in that particular case you will please note that
this is the semi-log paper in which the vertical axis is a linear scale is a linearly divided
scale and the horizontal axis is the log scale. So if you want log omega you have now to take
the log operation on omega, you simply insert your omega on this particular axis it will
appropriately take a point corresponding to log omega if linear scale were used. So it
is a non linear scale it is a log scale as you see and in this particular case the advantage
is this that every time log operation need not be taken. You have not to take the log
operation every time; you can directly take the value of omega in this particular thing.
Therefore, it appears that a question can come in anybody's mind that if one could take
the frequency this way so that the log of that frequency can automatically be taken
on this particular scale then why not take a log log paper so that the magnitude which
is available to me the log of that magnitude with a factor of 20 can be taken on this particular
scale hence this also could be a log scale where the real magnitude without taking the
log operation could be taken over here and automatically log of that will be taken and
db will be calculated.
This normally we do not do because the phase plot cannot be fitted into this particular
format. Phase plot is in absolute value. There since the phase plot is already additive you
are not going to take the log operation on the phase. Since the phase plot cannot appropriately
be fitted into the format of a log log paper it is most appropriate it is most convenient
as far as the Bode's plot study is concerned that we take a semi-log paper wherein the
vertical axis is your linear scale and the horizontal axis is the logarithmic scale.
So now onwards in my discussion when I take the horizontal axis please see I will be writing
omega all the time instead of writing log omega, hence that will automatically mean
I may not write again and again that the axis I am marking I am marking on a semi-log paper,
I will see to it that when I take these frequencies I appropriately make it non-linear and not
linear that non-linearity may not be true because it will be approximate however omega
verses db I may be showing on my plots here but they will definitely represent the axis
on a semi-log sheet. If that is the case..... now in this particular case please see that
I find the following building blocks.
If I know the magnitude and phase angle plots of these factors in that particular case I
can simply add up all of them to get the overall magnitude and phase angle. Please help me;
I have not taken a complete transfer function; if I miss any building block you have to help
me. I take the first building block as the magnitude K, the second building block I am
taking as j omega plus minus N general I am taking, it could represent a 0 at the origin
of certain order, it could represent and a pole at the origin though you know that practically
1 by j omega 1 by j omega squared and 1by j omega raise to the power of 0; type-1 type-0
type-1 type-2 systems are the most practical cases that we know but we can say that this
is the most general case.
Third case: (1 plus j omega T) plus minus m let us say this is a first-order factor
in the numerator or in the denominator, this can happen both ways; it can come in the numerator
as well as in the denominator. Fourth building block; please help me; the fourth building
block the general type of transfer functions which we have taken can I take this way? It
is (1 by omega n squared s squared plus 2 zeta by omega n s plus 1) raise to the power
of plus minus r even if you want to make it more general though multiple factors we hardly
come across. So it is a second-order factor and I have written it in this form which is
the time constant form, s I will replace as j omega. So this s is equal to j omega and
when I replace I will get the appropriate factor. It is a second-order factor which
can come both in the numerator and the denominator, this is another block which I can get, the
fourth building block.
Anything which I missed which you think you may come across as far as the Bode plot construction
is concerned, anything which I have missed please do help me? There could be something
let me list it here. Yes, I could list here probably the delay also: e to the power of
minus j omega tau d I told you. In process control systems the delay is quite frequently
available and therefore this in our models may come quite often and therefore let me
take this also as a building block: e to the power of minus j omega tau D where tau D is
the delay.
One more point please; let me not take a separate building block let me introduce here itself
and tell you that 1 minus j omega T or 1 over 1 minus j omega T is a possibility you cannot
rule out. What this corresponds to? This possibility will mean that the 0 of the system is in the
right-half lane; the system is stable but the 0 of the system is in the right-half plane,
this possibility simply means that the pole of the system is in the right-half plane.
I am not listing them as a separate building block because it is so easy to convert the
construction of this building block into this form because the magnitude will remain the
same the phase angle will simply get reversed, the minus sign will come. That is, the only
change that will come if instead of 1 plus j omega T I have the factor 1 minus j omega
T. So I will take up these building blocks quickly and then a composite example, use
of stability analysis using the Bode plot this is the thing which is left. I am sorry
we could complete it today so one or may be half more lecture on the subject next time,
thank you.