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Welcome to this ninth lecture of this course
and also microwave filter design So we are
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completing almost the design so in micro filter
design you see that upto this point we have
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seen the how to design RF filter with component
values lumped component values But you know
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that those lump component values lose their
values If you go to higher frequencies a capacitor
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can behave as an inductor an inductor can
be able and capacitor etc because those lumped
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elements they are not of reliable values at
high frequency
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Also it is difficult to make any value of
lump components at high frequencies Because
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here from the filter design you for a specific
insertion loss characteristic you are designing
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a particular value You are not sure whether
that value that that lump component is available
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or can be fabricated because there can be
fabrication difficulties But we know one thing
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that any lump component you can fabricate
by a transmission line by a shorted open transmission
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line you can make any impedance value because
we know that in a transmission line the input
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impedance that behaves as a either as a TAN
function or cotangent function So since TAN
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function extends from minus infinity to infinity
plus infinity and cotangent function also
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have minus infinity to plus infinity variation
So any value that means depending on the characteristic
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choosing the proper characteristic impedance
and the proper link we can design any value
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of inductance or capacitance at high frequency
that is done with the help of a transformation
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called Richard transformation . Will first
see P Richard if I remember correctly P Richard’s
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first introduce this transformation So they
make this microwave filters replacing the
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lamp component by stub transmission line Stubs
may be open stubs or short stubs as you all
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know you have all dealt with that you know
my microwave classes So with Richards transformation
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we can attempt what Richard transformation
says that you know that the input the impedance
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of any transmission line with characteristic
impedance Z0 and terminated by ZL That is
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given by ZL plus J Z0 tan beta L by Z0 plus
J ZL tan beta L all of you know this and we
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have also seen this we have proved this also
in the impedance transformer design we have
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extensively with these quarter wave transformer
we have fabricated from these etc Now you
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see that what Richard has done he is suppose
that let us this TAN Beta is an important
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thing So let us have a mapping that let us
define a capital gamma capital this gamma
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and that let us call it TAN beta L om so TAN
beta L we know in a transmission line what
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is this can be TAN Beta L where is the frequency
frequencies inside beta in transmission line
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a distributed transmission line the wave propagates
by TM mode in TM mode this beta can be written
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as omega by BP Okay so you see that by this
we are transforming the W plane to this gamma
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plane This capital gamma so now we assume
them that in the new plane this gamma is the
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angular frequency so we can write what happens
to inductors ZXL they will now will be calling
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previously we were calling J omega L now we
will be calling gamma L and that is what J
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you see instead of this I can say L is there
TAN beta L and in the susceptance PC that
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will be calling
So how we are getting this let us see from
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this equation this is his transformation this
this is the result that with this transformation
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this is called the Richards transformation
we get the new values are like this but let
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us see that what is this . so again I write
is Z in is Z0 plus J Z0 TAN beta L by Z0 plus
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J ZL TAN beta L now
when you know that we implement and inductor
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by a short circuited stub of electrical length
L where electrical length Beta L and we implement
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a capacitor by an open circuited stub of electrical
length Beta L so let us see those first that
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when we have shorted Stub shorted stub shorted
means load side we short that means Z1 is
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equal to 0 So what happens to input impedance
this is like this I have an Z0 I was this
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equation is for terminating with ZL now ZL
become zero so what is my input impedance
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this is my length L here the propagation constant
is beta so Zin will take the value Z0 then
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you see that J Z0 TAN beta L by Z0 so that
will be J Z0 TAN beta L Now you see Richard
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transformation shows that this Z for inductor
this is to be J L Tan beta L So I need to
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choose this Z0 as L that means to have a to
fabricated JXL that means an inductor with
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reactants XL I need to choose what this stub
it will be a shortage stub with L as the characteristic
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impedance and Length is L and its Beta is
given by that omega by BP So similarly so
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this part we have proved that how which had
got this similarly for let us do this thing
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again write . Zin is equal to Z0 ZL plus JZ0
Tan beta L by Z0 as JZL Tan beta L this time
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let us put an open circuit ZL is equal to
infinity and this is Z0 this is as before
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L and also the propagation constant is beta
so what happens what is Zin value under this
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condition Zin is Z0 this is infinity so divide
so you get 1 by J TAN beta L so that is 1
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by J TAN beta L Z0 then I can write Z1 is
capacitor inside Zbc that I can write as JC
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TAN Beta L so again it shows that I need to
choose this has Z0 value at C so that’s
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what he has done that you see he has chosen
a characteristic impedance of the line will
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be now chosen as C and length is L also aaah
for low pass prototype you see low pass prototype
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the cut off frequency was omega C is equal
to 1 so in Richard thing the mapping is to
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this omega plane so omega is capital omega
is now 1 So that is equal to TAN Beta L so
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this gives us what is the length L can I say
that this implies Beta L is PIE because TAN
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PIE by 4 is 1 so from that I can solve for
L is equal to PIE by 4 by what is beta beta
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is 2 PIE by lambda 2 PIE by lambda so I get
lambda by 8 so it says that all this lengths
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when we choose this that this is the length
L these all lengths at omega C is lambda by
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8 so at the cut off frequency the line so
I can say that . when you are doing Richards
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transformation this lambda is the wave length
at omega = omega C now obviously this means
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that at other frequencies these lines their
impedances is will change so lamb they wont
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represent the prototype lumped inductance
and capacitance properly but there is periodicity
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that after every after every I have omega
C then at five omega C again the element value
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will match this is because this beta L by
electrical length by beta L that is a periodic
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function of 2 PIE since am having lambda by
8 so I am having this variation that it is
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designed for omega C at other frequencies
it wont match but at 5 omega C again it will
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match okay So filter is function differ at
omega greater than omega C or omega less than
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omega C those value it will be different but
that you can tackle by other technics by broad
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banding etc that are advanced techniques so
now we can say that by Richard transformation
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in the low pass filter if there was a inductance
L that you can represent by in a distributed
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circuit by a short head thing you take the
characteristic impedance of the transmission
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line as L same value and this is lambda by
8 length at omega C similarly a capacitor
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of value see that with Richard transformation
again the length is same you see that is the
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beauty that all LC they will be of same length
but what is the value of Z0 this will be 1
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by C and you make this open open circuit this
is a short circuit now since all this length
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are of same length all the stubs are of same
length but they are of different variety either
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short circuit or open circuit depending on
it these lines are called commensurate lines
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So you need not bother about these line lengths
and if required suppose 1 length is not haaa
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very small or very large you can use this
periodicity and go to higher values so that
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higher or lower value so that you can get
that same thing so that means with this in
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the prototype design whenever this is a lumped
L or lumped C you can represent represent
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like this this so this is called as identity
but one more thing that is tat in high frequency
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another comes that . this lumped elements
suppose I have 1 L in a filter than a C now
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their distance is that also a matters because
there is phase difference between them when
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you are putting with this then that distances
etc may not be feasible and I need to sometimes
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I need to put some more gap or some less gap
so some redundancy redundant line needs to
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be incorporated So that these gaps etc they
are become feasible and there are sizable
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gaps between them some math sections need
to be put so more practical microfilter implementation
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requires these because sometimes I need to
physically separate transmission line stubs
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Then sometimes in fabrication if I have this
short circuit open circuit sometimes I need
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make for if there is a large circuit that
all the thing should be short circuit or all
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the things should be open circuit because
that makes fabrication easier but then I need
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to have transformation you know that can be
easily done because any haa aah suppose I
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have aah a implemented a inductor by a short
circuit now I can also implement that by an
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open circuit because the equations are from
I can have only the length etc they will have
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different values so sometimes we need to do
that that is why this is another need for
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practical design transform series stubs to
short stubs or vice-versa that means sometimes
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opposite some short stubs may be needed to
change to series stubs then sometimes change
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impractical characteristic impedances into
more realizable one Now these 3 are the practical
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things so motivation for using another technique
which is called Kuroda’s identity . Kuroda’s
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identity now kuroda he gave these things are
4 kuroda’s identities but he did that suppose
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I
have a lump filter I want to convert it to
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an inductor that means basically these shunt
open circuit I want to make as a series short
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circuit stub This is so what he did he said
with this you add let us say this was of Impedance
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1 by z2 he said add a Z1 this in kuroda’s
nomenclature it is called unit element these
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are redundant element is putting and showing
that this is equivalent to putting Z2 by N
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square some line with characteristic impedance
Z2 by N square and then in series with a inductor
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of characteristic impedance in N square similarly
this is called is first type kuroda’s identity
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first type kuroda’s identity second type
is like this that you have a series inductor
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this is equivalent to pieces that you have
this value is Z1 so here is saying this is
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equivalent to 1 by N square Z2 this one so
this is second type then third type is you
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have instead of in series inductor you have
a shunt inductor of value Z1 he said put with
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is a Z2 and this equivalent to Z2 by N square
then a you want to keep all inductor but y
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u need to have a transformer so this one is
saying Z1 by N square its characteristic impedance
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and this is 1 is to N square terms assured
transformer and is forth variety is
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you have it was a shunt you have a series
capacitance then you put Z2 ah sorry this
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is 1 by Z2 this is Z1 So this he says this
will be N square Z1 then you can written this
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1 by n Square Z2 but then with a transformer
whose ratio is M square is to M So in all
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the cases M square is equal to 1 plus Z2 by
Z1 so this each box represent and unit element
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unit element they are called unit element
basically it is same as a transmission line
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of unit unit element all these are unit elements
these
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are all what is unit element this is a transmission
line of length lambda 8 at omega C same as
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what we have seen the feature transformation
case and characteristic impedance in data
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transformation was ALRC here characteristic
impedance as indicated in the identities so
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Z1 is the characteristic impedance of these
box Z2 is the characteristic impedance Z2
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by N square is characteristic impedance etc
and lumped inductor and capacitor represents
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stubs of inductor represent the stubs of short
circuit and capacitor represent the stub of
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open circuit respectively now all this can
be proved We will just see how to prove this
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first identity let us say so first identity
. you see that I have basically you see here
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I have shunt capacitance so that means I have
a shunt stub open circuit stub and this value
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is 1 by Z2 characteristic impedance so these
stub 1 by Z2 Z2 and this length is lambda
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by 8 length this is a OC stub with this I
have a Z1 again the length all length you
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see are L which is lambda by 8 and this is
my unit element unit element now this if I
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have a transmission line of length L and characteristic
impedance Z1 I know it is ABCD matrix for
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this one this sorry one this particular one
ABCD matrix will be COS Beta L this we have
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done earlier that how to find ABCD matrix
so in earlier in NPTEL lectures you refer
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now in terms of Richard transformation I can
because there we have seen can be TAN beta
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L is equal to capital Omega so 1 by root over
1 plus Omega square into capital omega 1 J
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omega Z1 then J omega by Z1 and 1 just you
put then you get then this is remember capital
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omega is TAN beta L with this I can write
in this and also this open circuited stub
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what is the input impedance of this open circuited
stub that I can write as Z2 1 by J TAN Beta
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L and this is in terms of this minus JZ2 by
omega so now what is the this thing is nothing
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but like this Minus J Z2 by Omega So what
is it ABCD parameter ABCD will be simply 1
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0 is to 1 you see sorry ABCD C will be Z2
and this is 1 This you can check that this
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one already we have seen earlier . so now
the composite this whole thing composite thing
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ABCD matrix will be multiplication of these
two So I can write first I will have this
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is first that means this into this ABCD that
is the beauty these 2 are in cascade so I
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can easily write that 1 0 J omega by Z2 1
into 1J omega Z1 J omega by Z1 1 into that
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scalar multiplier 1 plus big omega square
So this 1 by root over 1 plus gamma square
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then 1 J Z1 J omega I by Z1 plus 1 by Z2 and
1 – omega square Z1 by Z2 this is the left
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hand side now right hand side if you look
at the first identity the first unit cell
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and then Z1 by N square so this I keep this
basically LHS I can say Left Hand Side ABCD
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matrix composite is this in . right hand side
I have this in its L length L characteristic
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impedance Z2 by N square then just look at
the figure this is L and this is Z1 by N square
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so this is unit element element sorry this
is short circuit series stub so I made it
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series stub so for unit element ABCD matrix
there will be 1 am sorry its square J omega
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N square by Z2 1 by root over 1 plus omega
square and this one short circuited stub again
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you see it is equivalent to this that J omega
Z1 By N Square series Stub so you know that
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ABCD matrix its ABCD will turn out to be 1
this thing we have done earlier just refer
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there so the composite so RHS will be composite
of this this is the first one in this one
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if you do that you get 1 by 1 plus Square
then 1 J omega by N square Z1 minus Z2 J omega
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N square by Z2 1 minus gamma square Z1 by
Z2 now you see this is RHS this is LHS so
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see LHS is these RHS is these so on this two
becomes identical only when I choose that
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in value that N square is equal to 1 plus
Z2 by Z1 thats why kuroda has done that and
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in all these cases this N square 1 + Z2 by
Z N so with this you can now have all the
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permutations given and so if you want to design
any a implementable filter upto Richards transformation
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you come then you take the appropriate kuroda’s
identity because you see which one you need
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because if you want to if you want need if
you have a shunt capacitance but you want
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to convert with series 1 you can have this
just add this unit element and in series with
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that you can get it similarly if you have
this you can use this this also popular these
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2 these 2 sometime you use but generally inductance
and they are not generally in this fashion
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but some band pass etc they are there but
low pass high pass generally we have this
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type of thing so you can use these 2 identities
but if you can refer to here and always do
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this so we will see some implementation of
these in the next lecture that how we go about
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these transformations Thank You