WEBVTT
Kind: captions
Language: en
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Welcome to this eighth lecture on microwave
filter design now we have seen how to design
00:00:28.509 --> 00:00:37.449
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
00:00:46.750 --> 00:00:57.830
is even now we also that time made a low pass
prototype with omega C the cutoff frequency
00:00:57.830 --> 00:01:04.430
is one hertz now we will see how to scale
this up so that topic is called as filter
00:01:04.430 --> 00:01:11.680
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
00:01:18.930 --> 00:01:27.280
one to the actual given cutoff frequency and
also we want to transform the nature of filter
00:01:27.280 --> 00:01:35.350
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
00:01:44.180 --> 00:01:52.500
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
00:02:04.340 --> 00:02:15.010
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
00:02:21.590 --> 00:02:30.270
that means source resistance from one ohm
to odd R0 ohm we are doing that time let the
00:02:30.270 --> 00:02:39.000
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
00:03:00.490 --> 00:03:17.900
it as R0 in the actual prototype then RS dashed
will be like this if it was RL in the original
00:03:17.900 --> 00:03:29.380
or in the prototype then after scaling it
will become R0 RL so this is impedance scaling
00:03:29.380 --> 00:03:40.410
so whenever we have impedance scaled from
one to R0 one ohm to R0 we have this R one
00:03:40.410 --> 00:03:48.670
one to RS we need to this so this is freq
impedance scaling then . we will the second
00:03:48.670 --> 00:04:12.290
variety the frequency scaling for low pass
filt ers So in omega C in prototype
00:04:12.290 --> 00:04:23.470
is one hertz but we need to change to some
other value omega C from one to omega we need
00:04:23.470 --> 00:04:33.790
to scale so that means you see if we need
to frequency dependence of the filter from
00:04:33.790 --> 00:04:42.509
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
00:04:50.979 --> 00:04:56.050
only it will be the cutoff frequency will
be change because cut off is one so you see
00:04:56.050 --> 00:05:01.740
then omega is equal to omega C this thing
again becomes one so that is the scaling that
00:05:01.740 --> 00:05:13.759
means if original insertion loss as a function
of omega was this we need to make that omega
00:05:13.759 --> 00:05:22.000
by omega C this will be then actually given
by the change quantity PLR dashed which is
00:05:22.000 --> 00:05:35.009
PLR with new omega so this is the scaling
so here this omega C is the new cut off frequency
00:05:35.009 --> 00:05:41.639
So when we do this that means when we make
from omega prototype we change by omega by
00:05:41.639 --> 00:05:53.199
omega C all the impedances they also change
so let us say that now the reactive impedance
00:05:53.199 --> 00:06:02.869
ZXK that will now be called Z omega by omega
C original LK so this in the new nomenclature
00:06:02.869 --> 00:06:13.389
will say J omega LK dashed after transformation
so the new value of LK dashed is nothing but
00:06:13.389 --> 00:06:23.180
LK just to compare here LK by omega C so that
means when we change the cutoff frequency
00:06:23.180 --> 00:06:32.719
all prototypes LK they should be divided by
omega C o get the new LK dashed values similarly
00:06:32.719 --> 00:06:46.339
we know that if we have the susceptance JBK
then that is J omeaga by omega C CK so that
00:06:46.339 --> 00:06:54.590
is J omega CK dashed from here if we compare
what will be CK dashed CK dashed will be CK
00:06:54.590 --> 00:07:02.430
by omega C so you see that all capacitances
of the prototype they need to be divided by
00:07:02.430 --> 00:07:12.300
omega C to get this now when we combine the
impedance scaling as well as the frequency
00:07:12.300 --> 00:07:34.199
scaling so combining combining frequency scaling
and impedance scaling will get LK dashed is
00:07:34.199 --> 00:07:49.520
R0 LK b omega C and CK dashed is equal to
CK by R0 omega C the reason is you can see
00:07:49.520 --> 00:07:56.979
that in case of impedance scaling the L dashed
is to be multiplied by R0 and C dashed is
00:07:56.979 --> 00:08:05.689
to be divided by R0 here for frequency scaling
both LK and Ck to be divided so here you see
00:08:05.689 --> 00:08:14.689
both are divided by omega C by one is multiplied
by R0 another is divided by R0 So this is
00:08:14.689 --> 00:08:21.669
frequency scaling and impedance scaling . now
we see that how the past behavior changes
00:08:21.669 --> 00:08:39.560
low pass to high pass low pass to high pass
transformation out prototype is always low
00:08:39.560 --> 00:08:59.550
pass but let us see what was our prototype
insertion loss PLR versus omega that was one
00:08:59.550 --> 00:09:10.390
here minus one this is 0 this is some value
given by the PLR minimum that we can pass
00:09:10.390 --> 00:09:21.810
band in pass band that we can now if we want
to make this two a characteristic like this
00:09:21.810 --> 00:09:34.640
that instead of one I want to make it omega
C and minus omega C this is 0 this is the
00:09:34.640 --> 00:09:49.360
new you can say PLR dashed then what I need
to do you see that if I change this that we
00:09:49.360 --> 00:10:03.380
have already seen ahhh that if I make omega
change to one by omega by omega C then when
00:10:03.380 --> 00:10:12.330
omega is equal to omega C I have one so this
I point is being mapped to omega C when omega
00:10:12.330 --> 00:10:20.190
is equal to minus omega C this point is mapped
to here when omega is equal to 0 the omeee
00:10:20.190 --> 00:10:27.740
this point this point maps to this point so
this is simply frequency scaling as we have
00:10:27.740 --> 00:10:36.121
already seen now we want to change the past
behavior that means what we want high pass
00:10:36.121 --> 00:10:48.220
means the PLR should be like this that this
our omega this is a new PLR you can say double
00:10:48.220 --> 00:10:58.390
dashed for high pass this was still low passed
with a changed omega but this is high passed
00:10:58.390 --> 00:11:09.180
here I want that suppose this is my level
tolerable getting in pass band so then I want
00:11:09.180 --> 00:11:18.070
this is my omega C and this is my minus omega
C so I want that at zero value the PLR should
00:11:18.070 --> 00:11:24.670
go to infinite at omega C PLR will come to
certain value and thereafter fall Similarly
00:11:24.670 --> 00:11:31.420
at minus omega and thereafter it should fall
that means I need a frequency variation inverse
00:11:31.420 --> 00:11:39.710
to this type of thing so we map instead of
this the new map sorry here you write the
00:11:39.710 --> 00:11:45.600
low pass to high pass transformation is obtained
omega that here I instead of omega that here
00:11:45.600 --> 00:11:53.660
I instead of this omega comes down and this
omega C comes up but also when I do this actually
00:11:53.660 --> 00:12:00.860
you know that high pass to low pass means
the inductance inductor and capacitor they
00:12:00.860 --> 00:12:07.800
will be interchanged now to have them in but
all the inductance and capacitances values
00:12:07.800 --> 00:12:15.450
L and C they need to be positive So to do
that I need to add a negative here so minus
00:12:15.450 --> 00:12:22.680
omega is changed to minus omega by omega C
this becomes a high pass filter you see this
00:12:22.680 --> 00:12:31.750
is high pass at lower value it is cutting
but it is giving so now we can find out what
00:12:31.750 --> 00:12:38.320
are the change in reactants and values inductor
and capacitors components values So again
00:12:38.320 --> 00:12:54.810
JXK which I will now write of J of minus omega
C this one minus omega C by omega LK and that
00:12:54.810 --> 00:13:01.260
I will give a name so it is obvious that omega
has come here so this has become a capacitance
00:13:01.260 --> 00:13:08.730
so I will call it one by J omega CK dashed
the new quantity CK dashed so from here you
00:13:08.730 --> 00:13:16.920
can see what is CK dashed CK dashed is nothing
but if you compare this CK dashed is one by
00:13:16.920 --> 00:13:28.260
omega C LK So the previous prototype LK that
has changed to CK dashed similarly I can do
00:13:28.260 --> 00:13:41.280
if susceptance of the capacitor JBK that will
be J minus omega C by omega into CK and that
00:13:41.280 --> 00:13:54.160
is one by here you see J omega LK dashed so
by comparing I now find the LK dashed is one
00:13:54.160 --> 00:14:02.370
by omega C by CK dashed You can easily check
that if I have not taken this minus then always
00:14:02.370 --> 00:14:08.610
this LK dashed and CK dashed wont be negative
so that will create an wont be positive that
00:14:08.610 --> 00:14:13.660
will create problem Because CK dashed LK dashed
components values they always need to be a
00:14:13.660 --> 00:14:19.630
positive that corresponding reactances may
become negative etc but this component values
00:14:19.630 --> 00:14:26.420
make it realizable we need to add this minus
sign now if we include here already we have
00:14:26.420 --> 00:14:32.420
included frequency scaling as well as pass
transformation We can include already we have
00:14:32.420 --> 00:14:40.050
seen the impedance scaling also so with impedance
scaling so that means I can say that impedance
00:14:40.050 --> 00:14:54.520
scaling class frequency scaling that means
from one to omega C plus low pass or LP two
00:14:54.520 --> 00:15:06.730
HP all this put together then my new CK dash
value is nothing but one by R0 omega C LK
00:15:06.730 --> 00:15:19.300
and LK dashed value is R0 by omega C Ck so
a inductor changes to CK dashed when I make
00:15:19.300 --> 00:15:26.870
this is well known but with this scaling you
need to need to know that if I want to make
00:15:26.870 --> 00:15:34.610
from a low pass prototype to a high pass filter
the inductance in the low pass filter prototype
00:15:34.610 --> 00:15:41.850
will be changed to a capacitor with this value
and the capacitor in a low pass filter will
00:15:41.850 --> 00:15:50.880
be changed to inductor with these value so
in tutorials we will see problems with this
00:15:50.880 --> 00:16:06.750
and now this is low to high pass transformation
we now see that what happen to low pass to
00:16:06.750 --> 00:16:28.860
band pass transformation
so originally again let me draw the PLR of
00:16:28.860 --> 00:16:42.500
the prototype this was the original PLR this
is omega this is 0 this is plus one this is
00:16:42.500 --> 00:17:01.310
minus one this was the pass band now what
I want as a band pass I want that it should
00:17:01.310 --> 00:17:15.419
be
this is my PLR new PLR I am writing same omega
00:17:15.419 --> 00:17:25.369
then you see I want that it should go to omega
0 and also there will be a high frequency
00:17:25.369 --> 00:17:33.539
upto which it will pass let us call this omega
one similarly here it will minus omega 0 this
00:17:33.539 --> 00:17:49.149
is the minus omega one this is minus omega
two anything I have left ok ok so this is
00:17:49.149 --> 00:18:00.049
thing now for this we want to find out what
is omega one omega one is the edges of the
00:18:00.049 --> 00:18:15.970
pass band so I can write omega two is the
upper edge of pass band omega one is the lower
00:18:15.970 --> 00:18:29.490
edge of pass band and omega 0 now omega 0
we can take either as amplitube hah arithmetic
00:18:29.490 --> 00:18:36.289
mean of omega one and omega two or we can
take it as geometric mean of omega two now
00:18:36.289 --> 00:18:42.039
both possibilities are there But equation
transformation will become easier if we take
00:18:42.039 --> 00:18:47.809
it at geometric mean that is why we generally
take it as omega 0 is equal to omega one omega
00:18:47.809 --> 00:18:57.549
two if we take that and the transformation
we call that omega of prototype should be
00:18:57.549 --> 00:19:08.019
changed like this omega 0 by omega two minus
omega one into one omega minus omega 0 by
00:19:08.019 --> 00:19:15.610
omega 0 by omega now we can ask how i got
this ok before that let me simplify this bit
00:19:15.610 --> 00:19:23.341
let us also called that this term let us call
delta this is no omega term here this is a
00:19:23.341 --> 00:19:31.009
constant so let me denote it by a constant
delta is equal to omega two minus omega one
00:19:31.009 --> 00:19:40.499
by omega 0 you know what is this? this is
nothing but delta is nothing but the fractional
00:19:40.499 --> 00:19:49.220
band width because band with is F two minus
F one everything is multiplied by one PIE
00:19:49.220 --> 00:19:55.059
here or this is angler frequency but when
we take the ratio it is same as fractional
00:19:55.059 --> 00:20:03.380
bandwidth now F two minus F one by F0 so this
delta is this so in terms of that side I can
00:20:03.380 --> 00:20:14.889
write one by delta into omega by omega 0 minus
omega 0 by omega now you see that I want that
00:20:14.889 --> 00:20:21.370
this point the minimum attenuation point in
the minimum insertion loss point in the pass
00:20:21.370 --> 00:20:30.519
band that should be mapped to omega 0 so you
see when omega is equal to 0 if I put then
00:20:30.519 --> 00:20:46.590
this is 0 ahhhhhh sorry you want I want that
omega is equal to 0 or actually here it is
00:20:46.590 --> 00:20:53.749
actually scaled version of this so omega 0
when it is omega is equal to omega 0 this
00:20:53.749 --> 00:21:01.139
point should map here you see omega is equal
to omega 0 so let me put or let me put it
00:21:01.139 --> 00:21:08.790
here here that this is my transformation keep
it what omega is omega 0 what happens to this
00:21:08.790 --> 00:21:22.019
new term one by delta then omega 0 by omega
0 minus omega 0 by omega 0 so it is 0 so I
00:21:22.019 --> 00:21:30.840
can say that this minimum pass band point
maps to this omega 0 point here now when omega
00:21:30.840 --> 00:21:50.200
is because I can say that minimum pass band
attenuation point maps to omega 0 when omega
00:21:50.200 --> 00:22:00.299
is equal to omega one what happens one by
delta then omega one by omega 0 minus omega
00:22:00.299 --> 00:22:15.480
0 by omega one so this is one by delta omega
one square sorry omega one square minus omega
00:22:15.480 --> 00:22:29.940
0 square by omega 0 omega one then you can
find out that what is omega 0 square you see
00:22:29.940 --> 00:22:38.759
already we have put this so one by delta omega
one square minus omega one omega two by omega
00:22:38.759 --> 00:22:49.100
0 omega one now that you know that you know
one by delta so I can omega minus one omega
00:22:49.100 --> 00:22:56.210
two by omega 0 and what is delta again you
see delta is omega two minus omega one by
00:22:56.210 --> 00:23:05.139
omega 0 so can I say that this will be minus
one that means it says that the left pass
00:23:05.139 --> 00:23:12.740
band edge I have put here omega is equal to
omega one so the left pass band edge that
00:23:12.740 --> 00:23:23.940
will mapped to the omega one that means this
point mapped to omega one so let me write
00:23:23.940 --> 00:23:38.879
the effect of this the left pass band edge
maps to omega one similarly you can do similarly
00:23:38.879 --> 00:23:48.360
that if I put omega is equal to two then the
whole thing one by delta omega two by omega
00:23:48.360 --> 00:24:00.070
0 minus omega 0 by omega two that becomes
plus one so I can say that right pass band
00:24:00.070 --> 00:24:12.399
edge maps to omega two and you get a graph
like this so this is exactly here similarly
00:24:12.399 --> 00:24:21.350
you can also find out that if I make now this
completes this part now you see . if we map
00:24:21.350 --> 00:24:33.259
again omega is equal to minus omega 0 you
will see that minimum pass band attenuation
00:24:33.259 --> 00:24:50.450
again maps to minus omega 0 minimum pass band
attenuation point maps to minus omega 0 if
00:24:50.450 --> 00:25:02.710
you take omega is equal to omega one then
you see that the if you maps to minus omega
00:25:02.710 --> 00:25:09.419
one then right pass band edge instead of left
pass band edge for positive frequency Right
00:25:09.419 --> 00:25:19.350
pass band edge maps to minus omega one omega
is equal to minus omega two you will see left
00:25:19.350 --> 00:25:30.330
pass band edge maps to minus omega two So
you get a graph or plot like this so transformation
00:25:30.330 --> 00:25:39.159
is correct now we will have to find out what
is the corresponding component value you see
00:25:39.159 --> 00:25:49.919
so with that transformation now an inductor
will be like this J by delta Omega by omega
00:25:49.919 --> 00:26:04.230
0 minus omega 0 by omega into LK so that will
give you j omega LK dashed minus J by omega
00:26:04.230 --> 00:26:19.330
CK dashed so basically you see a series inductor
becomes a series LC circuit with a new value
00:26:19.330 --> 00:26:29.242
of inductor and new value of capacitor So
you can see that if in the originally I have
00:26:29.242 --> 00:26:39.860
a LK now I am having or now I am having an
L now that is after transformation that is
00:26:39.860 --> 00:26:53.749
becoming a LK dashed and CK dashed in series
and what is the value of CK dashed if you
00:26:53.749 --> 00:27:06.669
just compare LK dashed is nothing but LK by
delta L0 or let me call LK itself So LK dashed
00:27:06.669 --> 00:27:16.659
is LK by delta omega 0 delta into omega 0
and CK dashed that is equal to delta by omega
00:27:16.659 --> 00:27:28.539
0 LK . similarly the capacitor in the prototype
so JBK is susceptance that will become now
00:27:28.539 --> 00:27:41.710
J by delta omega by omega 0 minus omega 0
by omega CK so that is J omega CK dashed minus
00:27:41.710 --> 00:27:53.100
J by omega LK dashed so a you see that here
a shunt capacitor transform to a shunt LC
00:27:53.100 --> 00:28:00.769
circuit with elements LK dashed and CK dashed
so if I draw again I had a CK dashed so if
00:28:00.769 --> 00:28:15.419
I draw again a low pass prototype but now
am having a shunt LC with LK dashed and CK
00:28:15.419 --> 00:28:23.789
dashed What are their value LK dashed just
compare here you will get delta dash is delta
00:28:23.789 --> 00:28:38.710
by omega 0 CK CK dashed is equal to CK by
delta omega 0 so also you can check that both
00:28:38.710 --> 00:28:45.220
these series resonant circuit and anti parallel
resonant circuit they their resonance frequencies
00:28:45.220 --> 00:28:51.139
as omega 0 so at omega 0 you see that omega
0 you see that whole circuit resonance that
00:28:51.139 --> 00:29:01.690
is why PLR you get as 0 so this is the band
pass transformation now we want to just I
00:29:01.690 --> 00:29:11.169
am giving the band stop transformation . in
band stop
00:29:11.169 --> 00:29:19.720
we have omega just inverse of these they are
omega delta was one by delta here omega delta
00:29:19.720 --> 00:29:29.359
is this omega by omega 0 minus omega 0 by
omega to the power minus one so you can just
00:29:29.359 --> 00:29:36.850
check that you know band pass characteristic
that will be something like this that instead
00:29:36.850 --> 00:29:45.470
of this you will have that at this point this
will be high so just inverse of this just
00:29:45.470 --> 00:29:58.039
you can check it and there if you have an
LK that will be changed to a parallel resonance
00:29:58.039 --> 00:30:12.809
circuit LK with LK dashed with CK dashed Where
LK dashed will be delta LK by omega 0 CK dashed
00:30:12.809 --> 00:30:27.799
is equal to one by omega 0 delta LK and a
capacitance CK that will be given by a series
00:30:27.799 --> 00:30:37.750
resonance circuit LK dashed CK dashed where
LK dashed is equal to one by omega 0 Delta
00:30:37.750 --> 00:30:50.080
CK and CK dashed is equal to delta CK by omega
0 that is all So you now have high pass band
00:30:50.080 --> 00:30:58.100
band stop anything you want to band pass you
can do so with this now we were in a position
00:30:58.100 --> 00:31:07.190
to find out what is a ahh actual filter specification
If it is a band stop or if it is a high high
00:31:07.190 --> 00:31:14.619
pass then from the prototype we can design
it but then there will be some issues with
00:31:14.619 --> 00:31:19.649
high frequency implementation of microwave
implementation of that will take up in next
00:31:19.649 --> 00:31:21.469
class Thank you