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Welcome to this eighth lecture on microwave
filter design now we have seen how to design
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various low pass prototype filters and that
time we have assumed in the prototype that
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source and load resistances they are unity
except in when in Chebyshav filter when n
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is even now we also that time made a low pass
prototype with omega C the cutoff frequency
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is one hertz now we will see how to scale
this up so that topic is called as filter
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transformation now one transformation is impedance
we need to change to the actual source and
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load resistances and also in frequency we
need to transform from omega C is equal to
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one to the actual given cutoff frequency and
also we want to transform the nature of filter
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from low pass to either high pass or band
pass or band stop so one by one will see this
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three types of transformation the first one
is called impedance scaling when we want to
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change the source and load resistances to
the actual given values specified values a
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so a source resistance RO suppose if the source
resistance is R0 then I can be multiplied
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by multiplying the impedance of the prototype
by this value R0 so by this all the maybe
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reactive may be resistive they needs to be
scaled up so when the multiplier in the impedance
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that means source resistance from one ohm
to odd R0 ohm we are doing that time let the
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in after impedance scaling the impedance is
becoming prime quantities so a L dashed suppose
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in low pass prototype it was L so after impedance
scaling it will become L dashed so the multiplier
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will be R0 L similarly if originally it was
C then here it will be C by R0 similarly if
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it as R0 in the actual prototype then RS dashed
will be like this if it was RL in the original
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or in the prototype then after scaling it
will become R0 RL so this is impedance scaling
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so whenever we have impedance scaled from
one to R0 one ohm to R0 we have this R one
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one to RS we need to this so this is freq
impedance scaling then . we will the second
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variety the frequency scaling for low pass
filt ers So in omega C in prototype
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is one hertz but we need to change to some
other value omega C from one to omega we need
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to scale so that means you see if we need
to frequency dependence of the filter from
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omega we need to change by omega by omega
C So we need to replace the omega in the prototype
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this is the prototype in the prototype thing
that to be scaled by omega by omega C then
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only it will be the cutoff frequency will
be change because cut off is one so you see
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then omega is equal to omega C this thing
again becomes one so that is the scaling that
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means if original insertion loss as a function
of omega was this we need to make that omega
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by omega C this will be then actually given
by the change quantity PLR dashed which is
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PLR with new omega so this is the scaling
so here this omega C is the new cut off frequency
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So when we do this that means when we make
from omega prototype we change by omega by
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omega C all the impedances they also change
so let us say that now the reactive impedance
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ZXK that will now be called Z omega by omega
C original LK so this in the new nomenclature
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will say J omega LK dashed after transformation
so the new value of LK dashed is nothing but
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LK just to compare here LK by omega C so that
means when we change the cutoff frequency
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all prototypes LK they should be divided by
omega C o get the new LK dashed values similarly
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we know that if we have the susceptance JBK
then that is J omeaga by omega C CK so that
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is J omega CK dashed from here if we compare
what will be CK dashed CK dashed will be CK
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by omega C so you see that all capacitances
of the prototype they need to be divided by
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omega C to get this now when we combine the
impedance scaling as well as the frequency
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scaling so combining combining frequency scaling
and impedance scaling will get LK dashed is
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R0 LK b omega C and CK dashed is equal to
CK by R0 omega C the reason is you can see
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that in case of impedance scaling the L dashed
is to be multiplied by R0 and C dashed is
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to be divided by R0 here for frequency scaling
both LK and Ck to be divided so here you see
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both are divided by omega C by one is multiplied
by R0 another is divided by R0 So this is
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frequency scaling and impedance scaling . now
we see that how the past behavior changes
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low pass to high pass low pass to high pass
transformation out prototype is always low
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pass but let us see what was our prototype
insertion loss PLR versus omega that was one
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here minus one this is 0 this is some value
given by the PLR minimum that we can pass
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band in pass band that we can now if we want
to make this two a characteristic like this
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that instead of one I want to make it omega
C and minus omega C this is 0 this is the
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new you can say PLR dashed then what I need
to do you see that if I change this that we
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have already seen ahhh that if I make omega
change to one by omega by omega C then when
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omega is equal to omega C I have one so this
I point is being mapped to omega C when omega
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is equal to minus omega C this point is mapped
to here when omega is equal to 0 the omeee
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this point this point maps to this point so
this is simply frequency scaling as we have
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already seen now we want to change the past
behavior that means what we want high pass
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means the PLR should be like this that this
our omega this is a new PLR you can say double
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dashed for high pass this was still low passed
with a changed omega but this is high passed
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here I want that suppose this is my level
tolerable getting in pass band so then I want
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this is my omega C and this is my minus omega
C so I want that at zero value the PLR should
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go to infinite at omega C PLR will come to
certain value and thereafter fall Similarly
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at minus omega and thereafter it should fall
that means I need a frequency variation inverse
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to this type of thing so we map instead of
this the new map sorry here you write the
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low pass to high pass transformation is obtained
omega that here I instead of omega that here
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I instead of this omega comes down and this
omega C comes up but also when I do this actually
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you know that high pass to low pass means
the inductance inductor and capacitor they
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will be interchanged now to have them in but
all the inductance and capacitances values
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L and C they need to be positive So to do
that I need to add a negative here so minus
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omega is changed to minus omega by omega C
this becomes a high pass filter you see this
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is high pass at lower value it is cutting
but it is giving so now we can find out what
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are the change in reactants and values inductor
and capacitors components values So again
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JXK which I will now write of J of minus omega
C this one minus omega C by omega LK and that
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I will give a name so it is obvious that omega
has come here so this has become a capacitance
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so I will call it one by J omega CK dashed
the new quantity CK dashed so from here you
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can see what is CK dashed CK dashed is nothing
but if you compare this CK dashed is one by
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omega C LK So the previous prototype LK that
has changed to CK dashed similarly I can do
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if susceptance of the capacitor JBK that will
be J minus omega C by omega into CK and that
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is one by here you see J omega LK dashed so
by comparing I now find the LK dashed is one
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by omega C by CK dashed You can easily check
that if I have not taken this minus then always
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this LK dashed and CK dashed wont be negative
so that will create an wont be positive that
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will create problem Because CK dashed LK dashed
components values they always need to be a
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positive that corresponding reactances may
become negative etc but this component values
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make it realizable we need to add this minus
sign now if we include here already we have
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included frequency scaling as well as pass
transformation We can include already we have
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seen the impedance scaling also so with impedance
scaling so that means I can say that impedance
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scaling class frequency scaling that means
from one to omega C plus low pass or LP two
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HP all this put together then my new CK dash
value is nothing but one by R0 omega C LK
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and LK dashed value is R0 by omega C Ck so
a inductor changes to CK dashed when I make
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this is well known but with this scaling you
need to need to know that if I want to make
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from a low pass prototype to a high pass filter
the inductance in the low pass filter prototype
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will be changed to a capacitor with this value
and the capacitor in a low pass filter will
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be changed to inductor with these value so
in tutorials we will see problems with this
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and now this is low to high pass transformation
we now see that what happen to low pass to
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band pass transformation
so originally again let me draw the PLR of
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the prototype this was the original PLR this
is omega this is 0 this is plus one this is
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minus one this was the pass band now what
I want as a band pass I want that it should
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be
this is my PLR new PLR I am writing same omega
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then you see I want that it should go to omega
0 and also there will be a high frequency
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upto which it will pass let us call this omega
one similarly here it will minus omega 0 this
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is the minus omega one this is minus omega
two anything I have left ok ok so this is
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thing now for this we want to find out what
is omega one omega one is the edges of the
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pass band so I can write omega two is the
upper edge of pass band omega one is the lower
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edge of pass band and omega 0 now omega 0
we can take either as amplitube ha uhhh arithmetic
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mean of omega one and omega two or we can
take it as geometric mean of omega two now
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both possibilities are there But equation
transformation will become easier if we take
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it at geometric mean that is why we generally
take it as omega 0 is equal to omega one omega
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two if we take that and the transformation
we call that omega of prototype should be
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changed like this omega 0 by omega two minus
omega one into one omega minus omega 0 by
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omega 0 by omega now we can ask how i got
this ok before that let me simplify this bit
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let us also called that this term let us call
delta this is no omega term here this is a
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constant so let me denote it by a constant
delta is equal to omega two minus omega one
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by omega 0 you know what is this? this is
nothing but delta is nothing but the fractional
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band width because band with is F two minus
F one everything is multiplied by one PIE
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here or this is angler frequency but when
we take the ratio it is same as fractional
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bandwidth now F two minus F one by F0 so this
delta is this so in terms of that side I can
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write one by delta into omega by omega 0 minus
omega 0 by omega now you see that I want that
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this point the minimum attenuation point in
the minimum insertion loss point in the pass
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band that should be mapped to omega 0 so you
see when omega is equal to 0 if I put then
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this is 0 ahhhhhh sorry you want I want that
omega is equal to 0 or actually here it is
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actually scaled version of this so omega 0
when it is omega is equal to omega 0 this
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point should map here you see omega is equal
to omega 0 so let me put or let me put it
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here here that this is my transformation keep
it what omega is omega 0 what happens to this
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new term one by delta then omega 0 by omega
0 minus omega 0 by omega 0 so it is 0 so I
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can say that this minimum pass band point
maps to this omega 0 point here now when omega
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is because I can say that minimum pass band
attenuation point maps to omega 0 when omega
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is equal to omega one what happens one by
delta then omega one by omega 0 minus omega
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0 by omega one so this is one by delta omega
one square sorry omega one square minus omega
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0 square by omega 0 omega one then you can
find out that what is omega 0 square you see
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already we have put this so one by delta omega
one square minus omega one omega two by omega
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0 omega one now that you know that you know
one by delta so I can omega minus one omega
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two by omega 0 and what is delta again you
see delta is omega two minus omega one by
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omega 0 so can I say that this will be minus
one that means it says that the left pass
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band edge I have put here omega is equal to
omega one so the left pass band edge that
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will mapped to the omega one that means this
point mapped to omega one so let me write
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the effect of this the left pass band edge
maps to omega one similarly you can do similarly
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that if I put omega is equal to two then the
whole thing one by delta omega two by omega
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0 minus omega 0 by omega two that becomes
plus one so I can say that right pass band
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edge maps to omega two and you get a graph
like this so this is exactly here similarly
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you can also find out that if I make now this
completes this part now you see . if we map
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again omega is equal to minus omega 0 you
will see that minimum pass band attenuation
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again maps to minus omega 0 minimum pass band
attenuation point maps to minus omega 0 if
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you take omega is equal to omega one then
you see that the if you maps to minus omega
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one then right pass band edge instead of left
pass band edge for positive frequency Right
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pass band edge maps to minus omega one omega
is equal to minus omega two you will see left
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pass band edge maps to minus omega two So
you get a graph or plot like this so transformation
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is correct now we will have to find out what
is the corresponding component value you see
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so with that transformation now an inductor
will be like this J by delta Omega by omega
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0 minus omega 0 by omega into LK so that will
give you j omega LK dashed minus J by omega
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CK dashed so basically you see a series inductor
becomes a series LC circuit with a new value
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of inductor and new value of capacitor So
you can see that if in the originally I have
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a LK now I am having or now I am having an
L now that is after transformation that is
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becoming a LK dashed and CK dashed in series
and what is the value of CK dashed if you
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just compare LK dashed is nothing but LK by
delta L0 or let me call LK itself So LK dashed
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is LK by delta omega 0 delta into omega 0
and CK dashed that is equal to delta by omega
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0 LK . similarly the capacitor in the prototype
so JBK is susceptance that will become now
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J by delta omega by omega 0 minus omega 0
by omega CK so that is J omega CK dashed minus
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J by omega LK dashed so a you see that here
a shunt capacitor transform to a shunt LC
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circuit with elements LK dashed and CK dashed
so if I draw again I had a CK dashed so if
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I draw again a low pass prototype but now
am having a shunt LC with LK dashed and CK
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dashed What are their value LK dashed just
compare here you will get delta dash is delta
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by omega 0 CK CK dashed is equal to CK by
delta omega 0 so also you can check that both
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these series resonant circuit and anti parallel
resonant circuit they their resonance frequencies
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as omega 0 so at omega 0 you see that omega
0 you see that whole circuit resonance that
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is why PLR you get as 0 so this is the band
pass transformation now we want to just I
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am giving the band stop transformation . in
band stop
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we have omega just inverse of these they are
omega delta was one by delta here omega delta
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is this omega by omega 0 minus omega 0 by
omega to the power minus one so you can just
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check that you know band pass characteristic
that will be something like this that instead
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of this you will have that at this point this
will be high so just inverse of this just
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you can check it and there if you have an
LK that will be changed to a parallel resonance
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circuit LK with LK dashed with CK dashed Where
LK dashed will be delta LK by omega 0 CK dashed
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is equal to one by omega 0 delta LK and a
capacitance CK that will be given by a series
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resonance circuit LK dashed CK dashed where
LK dashed is equal to one by omega 0 Delta
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CK and CK dashed is equal to delta CK by omega
0 that is all So you now have high pass band
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band stop anything you want to band pass you
can do so with this now we were in a position
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to find out what is a ahh actual filter specification
If it is a band stop or if it is a high high
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pass then from the prototype we can design
it but then there will be some issues with
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high frequency implementation of microwave
implementation of that will take up in next
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class Thank you