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Welcome to this 6 lecture of this lecture
series that today we will see ah in insertion
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loss based insertion loss based microwave
filter design Now already we have seen the
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motivation for going for this design in the
previous lecture Now today let me first define
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what is insertion loss? Insertion loss
is its symbol is PLR and this is ratio of
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power available from source to power delivered
to load by any subsystem Suppose I have any
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subsystem in this particular case it will
be filtered but insertion loss is a generalize
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concept that is why let me call it a electronic
network It is a black box it is 2 port black
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box now this side it will be connected
maybe to some other block but for this network
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this is the source side that so it will be
connected with this source Similarly it may
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be connected to some other block but to this
2 port network that is the load so it will
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be connected to another box which is called
load Now let the power source is delivering
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Pin and the load is taking PL So insertion
loss power available from source what is the
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power available from the source to these electronic
network that is Pin and so I can write this
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as Pin and power delivered to load is PL Obviously
you know that I do not know the impedance
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level of this so and I am assuming that in
general this is there will be some loss here
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also but in this particular case particularly
our filter case so this filter case there
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is no inside there is no power dissipation
because we will use only the reactive components
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and so the power is not dissipated inside
so it is a lossless part here internally But
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you know that due to that impedance mismatch
there can be reflections here so the wave
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is going and then the wave can come back so
there are reflections so all the power may
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not go here obviously the power that is reflected
that will go back here so here it will be
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something less power similarly here also due
to the impedance mismatch the wave is going
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like this but there may be reflections So
can I see that all this PN PL so overall reflection
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coefficient so can I say that this is equal
to 1 minus gamma which is a function of Omega
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This gamma is the reflection Coefficient over
all reflection coefficient which comprises
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of reflection here reflection coefficient
here so by that I can define an overall coefficient
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between this and that is gamma it is the function
of gam gamma and I know that the this ratio
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will be nothing but 1 minus this that means
1 minus reflected power this gamma please
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remember we have earlier discussed that this
gamma is a voltage reflection co efficient
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that is why we are taking this square thing
because we are taking of power now a if there
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is no mismatch here that means already impedance
matching has taken place between source and
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this our filter network and also between load
and our filter work then it will be like these
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so there wont be this reflection co-efficient
is zero there but remember there is one more
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thing that is this network what about the
power I gave yesterday also I said depending
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on the propagation constant in the pass band
it will flow but in general a portion of that
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will go here In high frequency or microwave
what is that suppose if I give some voltage
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how much voltage comes here that I can express
by a ratio called S parameter S21 S21 So I
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can see that under matched condition so this
is one and also I should then say that this
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is S21 square this S21 is this is port sec
2 of the electronic network this is port 1
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So actually insertion loss is 1 by this into
S21 Square So under matched condition both
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source and load matched then PLR is nothing
but S21 square here no there is a pit fall
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in measurements many times I see that in indian
engineer they say that what is insertion loss
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it is a S21 square Please remember that if
you have enforced the matching then only it
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is true otherwise you should also consider
the reflection taking place both here and
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here Many times when the measurement takes
place this impedance mismatch is not taken
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care and people forget to in incorporate this
part the reflection co efficient part ok so
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now I can say that I have define insertion
loss that means physically what is this it
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says that if this network was not there some
power was flowing to the load if I inin If
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i include this in the circuit the power will
change so this difference is basically the
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insertion loss So due to the insertion of
the this particular electronic network how
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much loss am getting in the circuit so in
general . this insertion loss in IL this is
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expressed in db so is remember that 10 log
10 PLR is the insertion loss in db now here
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I will haah take a property of any microwave
network you may not be familiar with this
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in graduate level we do not teach that but
in post graduate the microwave technology
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courses there we prove one point that gamma
omega for any two port network that should
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be always an even function of omega So what
does that mean that means always this is a
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property believe me or if you want to see
this is proved in any microwave engineering
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course so in any book particularly the our
recommended book this book microwave Engineering
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by David Pozar it is already recommended in
your course So in this book this is derived
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that always gamma is event function of Omega
that means gamma omega is equal to gamma of
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minus Omega we will utilize this property
now let me write what is gamma omega Gamma
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Omega is the impedance of the network either
this side or this side looking at input impedance
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or output impedance here minus Z0 by Z0 plus
if the characteristic impedance of the this
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filter section is Z0 then source then this
Z omega is the source impedance if I am talking
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of this reflection if I am talking of this
reflection then this Z omega is the load impedance
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and here also there is another property that
any impedance source load etc or the input
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impedance there the Z omega it can be retained
as r omega plus jx omega that is the part
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of this impedance and reactive part of impedance
and r omega is always again is an event function
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of omega and x omega is odd function of omega
This property are also true as I said that
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gamma omega is event function similarly this
is also true so we will write that ok this
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will break into that r omega minus Z0 minus
jx omega and r omega plus Z0 plus J omega
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now what is gamma of minus omega because I
want to test what is the nature of this PLR
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So we are going there that is insertion loss
has some specific functional characteristics
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with respective omega So that we are trying
to see so let me see what is this this is
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utilizing this properties that r Omega minus
Z omega sorry minus jz omega by r omega plus
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Z omega minus jx omega but we know that this
two should be equal So if I say that then
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basically basically what is this right side
can also retain as the complex conjugate of
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gamma omega You see that this is real so at
this minus this by this minus this so we saw
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that minus gamma minus omega is nothing but
gamma conjugate omega if we have this so we
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can say that what happens to this term.13:40)
gamma omega square because this is there in
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the insertion loss expression so gamma omega
square is gamma omega into gamma this is equal
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to gamma mega is equal to gamma minus omega
this is by definition and then we have just
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seen that this is nothing but gamma minus
omega So I can write it as gamma minus omega
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whole square because I can replace this with
or I can write like this gamma of minus omega
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and gamma of minus omega so that is gamma
minus omega whole square So I can say you
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see the property that gamma omega whole square
can I say is an event function of Omega so
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what does it mean that gamma omega square
when I will synthesizes as I said from the
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start that actually ee in this insertion loss
base method we specify the insertion loss
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we specify the insertion loss which is nothing
but attenuation part that how it will go that
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in pass band stop band etc For that we require
to have this but these says that this can
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be of that I event function of omega means
allot function components odd components are
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0 that means it can have only this a plus
b omega square plus c omega 4 plus d omega
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6 etc no omega 1 omega 3 like that terms So
we can say that this gamma omega square can
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be represented by two polynomials where m
and n are real polynomials in omega square
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m and n are real polynomial You see this is
reflection coefficient so all obviously this
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will be m omega square plus n omega square
because this is this plus something so what
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is then PLR PLR is 1 by 1minus m omega square
by m omega square plus n omega square and
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that this 1 plus m omega square by n omega
square So it is says that you can specify
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anything but I will be able to realize a filter
only this insertion loss is specified in this
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given form that means if you specify something
with omega etc then it won’t be realizable
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Now based on this specification there are
various choices So already I the student to
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refer to another NPTEL course basic building
blocks of Microwave Engineering
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in lecture 9 basic . building blocks no no
no basic basic tools of Microwave Engineering
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my course there in lecture 9 actually that
was in respective impedance matching and there
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we have seen in details the properties of
various polynomials which was used for synthesis
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that time it was impedance transformers but
those are also valid for filters so please
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brush up your knowledge of basic polynomial
functions like butterwork polynomial Chebyshav
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polynomial elliptic polynomial maximally flat
which is nothing but butterworth etc etc so
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here we start with butterworth polynomial
so if insertion loss is specified in the form
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of butterworth polynomial you know butterworth
is also called as maximally flat because given
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the order compare to any other polynomial
function it as suppose if it is ordered in
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upto N derivatives are all zero so that is
why called as derivative flat response So
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butterworth polynomial so this is the flattest
possible pass band so that means if we specify
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the insertion loss in the pass band by butterworth
filter Then we can say that we will get a
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very flat response which is desirable in the
pass band Obviously always zero cannot be
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achieved that attenuation constant zero but
will achieve a very flat pass band and by
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specifying a level we can say that ok my pass
band is I am not attenuating this all the
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frequency components in this band not more
than this amount So for a low pass filter
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the specification so low pass filter the specification
if we follow butterworth you know butterworth
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polynomial is given like this So already omega
square it is a function so you see that it
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is a function only omega square so it is realizable
from this property of M and N that ok I can
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realize it so let us try to realize this and
N is the order of filter that you know sorry
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I can make it capital N So N is the order
of the filter
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omega C is the cut of frequency low pass filter
cut of frequency will be lo low pass filter
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angler given by omega C Now pass band if you
see this pass band pass band extends from
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omega is equal to zero to omega is equal to
C and at the bandage that means suppose I
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am plotting omega and PLR oh sorry P L R then
zero this is omega C now from here it will
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be having some stop band so PLR will change
but what is the baah PLR at this point we
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can find from here so at omega is equal omega
C PLR is equal to 1 plus K square So I can
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specify now you know butter worth response
since we have taken like butterworth response
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is like this its maximally flat very flat
But in pass band I will have some attenuation
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But this the maximally flat one but I should
know that what is the maximum attenuation
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I am having so this value is what 1 plus K
square So suppose I want that ok no more than
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3 db attenuation or no more than 1 db attenuation
So I will put that at I know that at at omega
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is equal to omega C at cut off in the pass
band the maximum attenuation takes place here
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in the butterworth polynomial and that value
is K square from that I can always find K
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so for a example if we choose that this our
insertion loss maximum value is the PLR maximum
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at pass band I wont tolerate more than 3 db
so if this is 3 db what happens to K that
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1 plus k square ten log of this is 3 so from
that you can find out K is equal to 1 under
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this condition . So we can plot this PLR versus
omega by omega c normalize with respective
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omega C so I know the values will be like
this by 5 1 15 etc so this is point where
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cut off will takes place and
so what is the value at zero let us also see
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the at zero what is PLR ? this is 1 so PLR
is 1 here and PLR this value is 1 plus K square
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also what happens to this polynomial when
omega is . . into to the stop band that means
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Omega by omega C is the large number then
can I say that for omega greater than omega
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C I can say PLR is approximately K square
omega by omega C whole to the power 2n So
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what is the rate of increase of PLR in the
stop band from this I can say that rate of
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increase of PLR is how much it is 20 n db
part decade This is well known if you have
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this expression you can always say this so
we see that here at attenuation that alpha
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increase monotonically with frequency But
we know that it is at lower side of the pass
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band it is not much but at bandage that means
near cut off it is increasing but I can always
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specify that where it will be level fixed
lamp and this rate of interest is 20n db per
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decade so how much you want to achieve this
rate so that basically by that you select
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what is the order of this filter order means
you will have to find how many sections you
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need to put ok So third order fourth order
means you will have this now let us got to
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the another design that instead of this if
I say that in the low pass filter unlike this
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Chebyshav ok I want this rate to increase
further that I want this should be sharper
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than this I do not mind obviously you cannot
choose everything so I will say I want a very
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sharp price compare to butterworth here I
do not mind if instead of monotonically increasing
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the in pass band the insertion loss repels
that means I can have high low but do not
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cross this limit so that is possible Chebyshav
is sharpest one . or much sharper than our
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butter worth so I have this PLR here omega
by omega C let we fix my 1plus k square this
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is designer will choose and this is my omega
C so a butterworth polynomial it can be so
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let me first draw the maximally flat one this
is my butterworth maximally flat or butter
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worth now this is Chebyshav Chebyshav as various
depending on the order that chebyshav various
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but it repels whether there will be one repel
1 cycle or 2 cycle 3 cycle that depends on
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order but you see always they are confined
between this and this but so what is the advantage
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you see that here it was at monotonically
increasing here I have repel but I can specify
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that ok B within this limit so in pass band
it is never crossing the limit Once it is
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going away from pass brand it is much sharper
than butterworth that’s it is advantage
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so if you want your prime concern is after
pass band when I entering stop band I will
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have a very good cut off and very sharp cut
off You opt for this chebyshav shape polynomial
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so again you can determine the polynomial
by this K square and what is this rate of
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increase if we see the so in this case first
me let me write down the chebyshav polynomial
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from 1+ksquare Tn square omega by omega C
Now Tn is the Chebyshav polynomial of . . and
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we choose like this so for large N again I
can say that when omega is greater than omega
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C PLR is K square Tn square omega by omega
C Now at high value of omega by omega C Tn
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chebyshav polynomial it is approximated very
good approximation is this so this becomes
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K square Tn ah sorry K square into half 2x
y to the power N so this is K square by 4
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2 this is Tnx here I will write 2 omega by
omega C So 2 omega by omega C e to the power
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N This is Tnx so Tnx square . . so here you
now tell what is the slope I have came here
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because I want it is sharper slope so slope
is 20 db by decade 20 ndb so same as butterworth
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then you can say that how I am getting it
and what is the advantage at advantage is
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that always this value of Tn it is always
2 to the power 2N by 4 times insertion loss
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of any chebyshav filter is 2 to the power
2N by 4 times larger than butterworth Now
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obviously if you go for N is equal to 1 that
means the single section chebyshav then you
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do not have any advantage but the moment you
go for N is equal to 2 or N is equal to 3
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etc so N is equal to 2 means you have 2 to
the power 4 sixteen by 4 that means 4 times
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larger the butterworth always you will have
the value so even if the slope are not much
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different but you are getting more value of
insertion loss so your alpha is increased
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So then we will see there is another filter
called another very popular filter . particularly
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in CD ROM etc this is used this called elliptic
filter Now if you want you have seen that
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butter worth and chebyshav same rate of price
but elliptic is the maximum rate of price
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it can give but what it will you will suffer
that it has repel both in pass band and stop
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band so checbyshav you see it monotonically
increases once it is in the pass band it had
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repel in once it is ahh at attenuation of
stop band here it is in pass band it is repel
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but it does not have any repel elliptic have
repel in both so that means here it will be
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something like these how many it depends on
the order so it goes and then here again it
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repels but no problem if you specify that
ok atleast everywhere I want this attenuation
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in the stop band this is your stop band this
is your pass band So here as before you can
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specify that ok instead of pass band it is
beyond this and stop band attenuation should
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be like this then you go for elliptic and
there is another one sometimes in a filter
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if you have if your bass band signal or if
you want to possess the signal any bass band
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signal is not a single tone it as a sprayed
of frequency suppose when am talking talking
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up to 20 Kilohertz or roughly from 4 kilohertz
to 20 kilohertz there will be the voice sorry
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20 hertz to 20 kilohertz So you see this differ
different frequency if they are attenuated
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differently then there is problem in the reconstruction
So sometimes that means a linear system I
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want filter should act as a linear system
sometimes sometimes I can tolerate that is
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not a problem but in some application I cannot
tolerate that that time I want that linear
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phase filter Sorry this is linear phase filter
so linear phase means what that my phase should
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be specified like this 1plus P omega by omega
c whole to the power 2N so if we specify that
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P is a constant but I should have this type
of variation so this is a as you can guess
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but in phase am specifying a butterworth type
polynomial this is nothing but a butterworth
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type polynomial so this is the phase of the
voltage transfer function of the filter that
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should satisfy these under this condition
the group delay because that is the measure
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of whether I can tolerate or not As a group
when the bass band signal is moving so bass
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band signal or any RF signal with a frequency
spread that what is the group delay that means
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what is the maximum id id what is the maximum
delay between the maximum phasing delay and
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minimum phasing delay so that is given by
D pie by D omega and you see it trans out
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to be 1 plus b 2n plus1 omega by omega C whole
to the power 2N So this is a again you it
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is a maximally flat response so that may be
tolerable so if you use the linear phase filter
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you have this Like that there can be other
specification other polynomial still this
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is search is going on people are using various
newer newer functions from mathematics taking
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new functions and implementing that to get
a more desired attenuation characteristic
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etc So we have seen that how to do it so now
we will try to see how a microwave engineer
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will implement this because this mathematics
or this specifications is one thing but finally
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as a engineer we should implement that for
that we need to have some mechanism there
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is nothing fundamentally new this is all about
filters but unless and until we engineers
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plan how to design how to implement it we
do not consider our job finished scientist
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upto this they stop but we engineer will always
go on and try to make that ok if I want to
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make it and taste it I should know after this
what will happen that will see in the next
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class the implementation of this filters thank
you