WEBVTT
Kind: captions
Language: en
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Welcome to this ninth lecture of this course
and also microwave filter design So we are
00:00:32.430 --> 00:00:42.120
completing almost the design so in micro filter
design you see that upto this point we have
00:00:42.120 --> 00:00:51.290
seen the how to design RF filter with component
values lumped component values But you know
00:00:51.290 --> 00:00:56.809
that those lump component values lose their
values If you go to higher frequencies a capacitor
00:00:56.809 --> 00:01:03.390
can behave as an inductor an inductor can
be able and capacitor etc because those lumped
00:01:03.390 --> 00:01:08.760
elements they are not of reliable values at
high frequency
00:01:08.760 --> 00:01:14.290
Also it is difficult to make any value of
lump components at high frequencies Because
00:01:14.290 --> 00:01:21.340
here from the filter design you for a specific
insertion loss characteristic you are designing
00:01:21.340 --> 00:01:27.010
a particular value You are not sure whether
that value that that lump component is available
00:01:27.010 --> 00:01:32.900
or can be fabricated because there can be
fabrication difficulties But we know one thing
00:01:32.900 --> 00:01:39.790
that any lump component you can fabricate
by a transmission line by a shorted open transmission
00:01:39.790 --> 00:01:46.060
line you can make any impedance value because
we know that in a transmission line the input
00:01:46.060 --> 00:01:55.750
impedance that behaves as a either as a TAN
function or cotangent function So since TAN
00:01:55.750 --> 00:02:01.720
function extends from minus infinity to infinity
plus infinity and cotangent function also
00:02:01.720 --> 00:02:10.310
have minus infinity to plus infinity variation
So any value that means depending on the characteristic
00:02:10.310 --> 00:02:15.150
choosing the proper characteristic impedance
and the proper link we can design any value
00:02:15.150 --> 00:02:22.950
of inductance or capacitance at high frequency
that is done with the help of a transformation
00:02:22.950 --> 00:02:31.190
called Richard transformation . Will first
see P Richard if I remember correctly P Richard’s
00:02:31.190 --> 00:02:39.650
first introduce this transformation So they
make this microwave filters replacing the
00:02:39.650 --> 00:02:47.530
lamp component by stub transmission line Stubs
may be open stubs or short stubs as you all
00:02:47.530 --> 00:02:53.720
know you have all dealt with that you know
my microwave classes So with Richards transformation
00:02:53.720 --> 00:03:01.110
we can attempt what Richard transformation
says that you know that the input the impedance
00:03:01.110 --> 00:03:10.000
of any transmission line with characteristic
impedance Z0 and terminated by ZL That is
00:03:10.000 --> 00:03:24.760
given by ZL plus J Z0 tan beta L by Z0 plus
J ZL tan beta L all of you know this and we
00:03:24.760 --> 00:03:30.500
have also seen this we have proved this also
in the impedance transformer design we have
00:03:30.500 --> 00:03:37.410
extensively with these quarter wave transformer
we have fabricated from these etc Now you
00:03:37.410 --> 00:03:46.860
see that what Richard has done he is suppose
that let us this TAN Beta is an important
00:03:46.860 --> 00:04:00.590
thing So let us have a mapping that let us
define a capital gamma capital this gamma
00:04:00.590 --> 00:04:17.180
and that let us call it TAN beta L om so TAN
beta L we know in a transmission line what
00:04:17.180 --> 00:04:24.710
is this can be TAN Beta L where is the frequency
frequencies inside beta in transmission line
00:04:24.710 --> 00:04:31.620
a distributed transmission line the wave propagates
by TM mode in TM mode this beta can be written
00:04:31.620 --> 00:04:47.939
as omega by BP Okay so you see that by this
we are transforming the W plane to this gamma
00:04:47.939 --> 00:05:02.759
plane This capital gamma so now we assume
them that in the new plane this gamma is the
00:05:02.759 --> 00:05:15.360
angular frequency so we can write what happens
to inductors ZXL they will now will be calling
00:05:15.360 --> 00:05:27.749
previously we were calling J omega L now we
will be calling gamma L and that is what J
00:05:27.749 --> 00:05:39.969
you see instead of this I can say L is there
TAN beta L and in the susceptance PC that
00:05:39.969 --> 00:05:53.479
will be calling
So how we are getting this let us see from
00:05:53.479 --> 00:05:59.879
this equation this is his transformation this
this is the result that with this transformation
00:05:59.879 --> 00:06:05.310
this is called the Richards transformation
we get the new values are like this but let
00:06:05.310 --> 00:06:21.389
us see that what is this . so again I write
is Z in is Z0 plus J Z0 TAN beta L by Z0 plus
00:06:21.389 --> 00:06:37.719
J ZL TAN beta L now
when you know that we implement and inductor
00:06:37.719 --> 00:06:45.389
by a short circuited stub of electrical length
L where electrical length Beta L and we implement
00:06:45.389 --> 00:06:53.979
a capacitor by an open circuited stub of electrical
length Beta L so let us see those first that
00:06:53.979 --> 00:07:06.020
when we have shorted Stub shorted stub shorted
means load side we short that means Z1 is
00:07:06.020 --> 00:07:17.439
equal to 0 So what happens to input impedance
this is like this I have an Z0 I was this
00:07:17.439 --> 00:07:24.330
equation is for terminating with ZL now ZL
become zero so what is my input impedance
00:07:24.330 --> 00:07:33.051
this is my length L here the propagation constant
is beta so Zin will take the value Z0 then
00:07:33.051 --> 00:07:56.050
you see that J Z0 TAN beta L by Z0 so that
will be J Z0 TAN beta L Now you see Richard
00:07:56.050 --> 00:08:04.629
transformation shows that this Z for inductor
this is to be J L Tan beta L So I need to
00:08:04.629 --> 00:08:15.509
choose this Z0 as L that means to have a to
fabricated JXL that means an inductor with
00:08:15.509 --> 00:08:26.459
reactants XL I need to choose what this stub
it will be a shortage stub with L as the characteristic
00:08:26.459 --> 00:08:40.700
impedance and Length is L and its Beta is
given by that omega by BP So similarly so
00:08:40.700 --> 00:08:49.360
this part we have proved that how which had
got this similarly for let us do this thing
00:08:49.360 --> 00:09:06.710
again write . Zin is equal to Z0 ZL plus JZ0
Tan beta L by Z0 as JZL Tan beta L this time
00:09:06.710 --> 00:09:15.590
let us put an open circuit ZL is equal to
infinity and this is Z0 this is as before
00:09:15.590 --> 00:09:27.090
L and also the propagation constant is beta
so what happens what is Zin value under this
00:09:27.090 --> 00:09:43.570
condition Zin is Z0 this is infinity so divide
so you get 1 by J TAN beta L so that is 1
00:09:43.570 --> 00:10:03.490
by J TAN beta L Z0 then I can write Z1 is
capacitor inside Zbc that I can write as JC
00:10:03.490 --> 00:10:14.720
TAN Beta L so again it shows that I need to
choose this has Z0 value at C so that’s
00:10:14.720 --> 00:10:22.440
what he has done that you see he has chosen
a characteristic impedance of the line will
00:10:22.440 --> 00:10:40.620
be now chosen as C and length is L also aaah
for low pass prototype you see low pass prototype
00:10:40.620 --> 00:10:47.780
the cut off frequency was omega C is equal
to 1 so in Richard thing the mapping is to
00:10:47.780 --> 00:10:59.450
this omega plane so omega is capital omega
is now 1 So that is equal to TAN Beta L so
00:10:59.450 --> 00:11:08.270
this gives us what is the length L can I say
that this implies Beta L is PIE because TAN
00:11:08.270 --> 00:11:16.981
PIE by 4 is 1 so from that I can solve for
L is equal to PIE by 4 by what is beta beta
00:11:16.981 --> 00:11:27.930
is 2 PIE by lambda 2 PIE by lambda so I get
lambda by 8 so it says that all this lengths
00:11:27.930 --> 00:11:38.490
when we choose this that this is the length
L these all lengths at omega C is lambda by
00:11:38.490 --> 00:11:48.860
8 so at the cut off frequency the line so
I can say that . when you are doing Richards
00:11:48.860 --> 00:12:05.820
transformation this lambda is the wave length
at omega = omega C now obviously this means
00:12:05.820 --> 00:12:15.780
that at other frequencies these lines their
impedances is will change so lamb they wont
00:12:15.780 --> 00:12:23.530
represent the prototype lumped inductance
and capacitance properly but there is periodicity
00:12:23.530 --> 00:12:34.380
that after every after every I have omega
C then at five omega C again the element value
00:12:34.380 --> 00:12:41.610
will match this is because this beta L by
electrical length by beta L that is a periodic
00:12:41.610 --> 00:12:48.510
function of 2 PIE since am having lambda by
8 so I am having this variation that it is
00:12:48.510 --> 00:12:54.560
designed for omega C at other frequencies
it wont match but at 5 omega C again it will
00:12:54.560 --> 00:13:04.740
match okay So filter is function differ at
omega greater than omega C or omega less than
00:13:04.740 --> 00:13:11.130
omega C those value it will be different but
that you can tackle by other technics by broad
00:13:11.130 --> 00:13:18.620
banding etc that are advanced techniques so
now we can say that by Richard transformation
00:13:18.620 --> 00:13:28.530
in the low pass filter if there was a inductance
L that you can represent by in a distributed
00:13:28.530 --> 00:13:37.280
circuit by a short head thing you take the
characteristic impedance of the transmission
00:13:37.280 --> 00:13:47.850
line as L same value and this is lambda by
8 length at omega C similarly a capacitor
00:13:47.850 --> 00:13:55.010
of value see that with Richard transformation
again the length is same you see that is the
00:13:55.010 --> 00:14:03.870
beauty that all LC they will be of same length
but what is the value of Z0 this will be 1
00:14:03.870 --> 00:14:13.970
by C and you make this open open circuit this
is a short circuit now since all this length
00:14:13.970 --> 00:14:20.500
are of same length all the stubs are of same
length but they are of different variety either
00:14:20.500 --> 00:14:28.650
short circuit or open circuit depending on
it these lines are called commensurate lines
00:14:28.650 --> 00:14:37.600
So you need not bother about these line lengths
and if required suppose 1 length is not haaa
00:14:37.600 --> 00:14:45.730
very small or very large you can use this
periodicity and go to higher values so that
00:14:45.730 --> 00:14:53.390
higher or lower value so that you can get
that same thing so that means with this in
00:14:53.390 --> 00:15:00.370
the prototype design whenever this is a lumped
L or lumped C you can represent represent
00:15:00.370 --> 00:15:06.881
like this this so this is called as identity
but one more thing that is tat in high frequency
00:15:06.881 --> 00:15:14.010
another comes that . this lumped elements
suppose I have 1 L in a filter than a C now
00:15:14.010 --> 00:15:19.750
their distance is that also a matters because
there is phase difference between them when
00:15:19.750 --> 00:15:28.100
you are putting with this then that distances
etc may not be feasible and I need to sometimes
00:15:28.100 --> 00:15:34.980
I need to put some more gap or some less gap
so some redundancy redundant line needs to
00:15:34.980 --> 00:15:41.510
be incorporated So that these gaps etc they
are become feasible and there are sizable
00:15:41.510 --> 00:15:49.290
gaps between them some math sections need
to be put so more practical microfilter implementation
00:15:49.290 --> 00:16:12.089
requires these because sometimes I need to
physically separate transmission line stubs
00:16:12.089 --> 00:16:20.860
Then sometimes in fabrication if I have this
short circuit open circuit sometimes I need
00:16:20.860 --> 00:16:26.970
make for if there is a large circuit that
all the thing should be short circuit or all
00:16:26.970 --> 00:16:32.240
the things should be open circuit because
that makes fabrication easier but then I need
00:16:32.240 --> 00:16:37.630
to have transformation you know that can be
easily done because any haa aah suppose I
00:16:37.630 --> 00:16:43.670
have aah a implemented a inductor by a short
circuit now I can also implement that by an
00:16:43.670 --> 00:16:50.300
open circuit because the equations are from
I can have only the length etc they will have
00:16:50.300 --> 00:16:56.800
different values so sometimes we need to do
that that is why this is another need for
00:16:56.800 --> 00:17:11.520
practical design transform series stubs to
short stubs or vice-versa that means sometimes
00:17:11.520 --> 00:17:21.179
opposite some short stubs may be needed to
change to series stubs then sometimes change
00:17:21.179 --> 00:17:38.429
impractical characteristic impedances into
more realizable one Now these 3 are the practical
00:17:38.429 --> 00:17:57.299
things so motivation for using another technique
which is called Kuroda’s identity . Kuroda’s
00:17:57.299 --> 00:18:22.140
identity now kuroda he gave these things are
4 kuroda’s identities but he did that suppose
00:18:22.140 --> 00:18:29.570
I
have a lump filter I want to convert it to
00:18:29.570 --> 00:18:39.480
an inductor that means basically these shunt
open circuit I want to make as a series short
00:18:39.480 --> 00:18:51.950
circuit stub This is so what he did he said
with this you add let us say this was of Impedance
00:18:51.950 --> 00:19:00.779
1 by z2 he said add a Z1 this in kuroda’s
nomenclature it is called unit element these
00:19:00.779 --> 00:19:15.119
are redundant element is putting and showing
that this is equivalent to putting Z2 by N
00:19:15.119 --> 00:19:29.070
square some line with characteristic impedance
Z2 by N square and then in series with a inductor
00:19:29.070 --> 00:19:38.240
of characteristic impedance in N square similarly
this is called is first type kuroda’s identity
00:19:38.240 --> 00:19:52.900
first type kuroda’s identity second type
is like this that you have a series inductor
00:19:52.900 --> 00:20:15.749
this is equivalent to pieces that you have
this value is Z1 so here is saying this is
00:20:15.749 --> 00:20:30.739
equivalent to 1 by N square Z2 this one so
this is second type then third type is you
00:20:30.739 --> 00:20:41.600
have instead of in series inductor you have
a shunt inductor of value Z1 he said put with
00:20:41.600 --> 00:21:08.390
is a Z2 and this equivalent to Z2 by N square
then a you want to keep all inductor but y
00:21:08.390 --> 00:21:20.200
u need to have a transformer so this one is
saying Z1 by N square its characteristic impedance
00:21:20.200 --> 00:21:32.419
and this is 1 is to N square terms assured
transformer and is forth variety is
00:21:32.419 --> 00:21:47.739
you have it was a shunt you have a series
capacitance then you put Z2 sorry this
00:21:47.739 --> 00:22:08.470
is 1 by Z2 this is Z1 So this he says this
will be N square Z1 then you can written this
00:22:08.470 --> 00:22:23.220
1 by n Square Z2 but then with a transformer
whose ratio is M square is to M So in all
00:22:23.220 --> 00:22:36.490
the cases M square is equal to 1 plus Z2 by
Z1 so this each box represent and unit element
00:22:36.490 --> 00:22:48.600
unit element they are called unit element
basically it is same as a transmission line
00:22:48.600 --> 00:23:03.200
of unit unit element all these are unit elements
these
00:23:03.200 --> 00:23:20.399
are all what is unit element this is a transmission
line of length lambda 8 at omega C same as
00:23:20.399 --> 00:23:28.909
what we have seen the feature transformation
case and characteristic impedance in data
00:23:28.909 --> 00:23:37.840
transformation was ALRC here characteristic
impedance as indicated in the identities so
00:23:37.840 --> 00:23:43.320
Z1 is the characteristic impedance of these
box Z2 is the characteristic impedance Z2
00:23:43.320 --> 00:23:51.590
by N square is characteristic impedance etc
and lumped inductor and capacitor represents
00:23:51.590 --> 00:23:59.139
stubs of inductor represent the stubs of short
circuit and capacitor represent the stub of
00:23:59.139 --> 00:24:08.100
open circuit respectively now all this can
be proved We will just see how to prove this
00:24:08.100 --> 00:24:21.299
first identity let us say so first identity
. you see that I have basically you see here
00:24:21.299 --> 00:24:31.850
I have shunt capacitance so that means I have
a shunt stub open circuit stub and this value
00:24:31.850 --> 00:24:44.049
is 1 by Z2 characteristic impedance so these
stub 1 by Z2 Z2 and this length is lambda
00:24:44.049 --> 00:24:54.559
by 8 length this is a OC stub with this I
have a Z1 again the length all length you
00:24:54.559 --> 00:25:07.669
see are L which is lambda by 8 and this is
my unit element unit element now this if I
00:25:07.669 --> 00:25:17.100
have a transmission line of length L and characteristic
impedance Z1 I know it is ABCD matrix for
00:25:17.100 --> 00:25:28.830
this one this sorry one this particular one
ABCD matrix will be COS Beta L this we have
00:25:28.830 --> 00:25:35.529
done earlier that how to find ABCD matrix
so in earlier in NPTEL lectures you refer
00:25:35.529 --> 00:25:50.539
now in terms of Richard transformation I can
because there we have seen can be TAN beta
00:25:50.539 --> 00:26:03.360
L is equal to capital Omega so 1 by root over
1 plus Omega square into capital omega 1 J
00:26:03.360 --> 00:26:23.230
omega Z1 then J omega by Z1 and 1 just you
put then you get then this is remember capital
00:26:23.230 --> 00:26:30.940
omega is TAN beta L with this I can write
in this and also this open circuited stub
00:26:30.940 --> 00:26:40.379
what is the input impedance of this open circuited
stub that I can write as Z2 1 by J TAN Beta
00:26:40.379 --> 00:26:50.029
L and this is in terms of this minus JZ2 by
omega so now what is the this thing is nothing
00:26:50.029 --> 00:27:05.340
but like this Minus J Z2 by Omega So what
is it ABCD parameter ABCD will be simply 1
00:27:05.340 --> 00:27:19.330
0 is to 1 you see sorry ABCD C will be Z2
and this is 1 This you can check that this
00:27:19.330 --> 00:27:25.190
one already we have seen earlier . so now
the composite this whole thing composite thing
00:27:25.190 --> 00:27:32.160
ABCD matrix will be multiplication of these
two So I can write first I will have this
00:27:32.160 --> 00:27:37.799
is first that means this into this ABCD that
is the beauty these 2 are in cascade so I
00:27:37.799 --> 00:27:58.909
can easily write that 1 0 J omega by Z2 1
into 1J omega Z1 J omega by Z1 1 into that
00:27:58.909 --> 00:28:08.830
scalar multiplier 1 plus big omega square
So this 1 by root over 1 plus gamma square
00:28:08.830 --> 00:28:29.350
then 1 J Z1 J omega I by Z1 plus 1 by Z2 and
1 – omega square Z1 by Z2 this is the left
00:28:29.350 --> 00:28:34.989
hand side now right hand side if you look
at the first identity the first unit cell
00:28:34.989 --> 00:28:44.749
and then Z1 by N square so this I keep this
basically LHS I can say Left Hand Side ABCD
00:28:44.749 --> 00:28:59.490
matrix composite is this in . right hand side
I have this in its L length L characteristic
00:28:59.490 --> 00:29:09.529
impedance Z2 by N square then just look at
the figure this is L and this is Z1 by N square
00:29:09.529 --> 00:29:21.279
so this is unit element element sorry this
is short circuit series stub so I made it
00:29:21.279 --> 00:29:37.109
series stub so for unit element ABCD matrix
there will be 1 am sorry its square J omega
00:29:37.109 --> 00:29:50.970
N square by Z2 1 by root over 1 plus omega
square and this one short circuited stub again
00:29:50.970 --> 00:30:05.609
you see it is equivalent to this that J omega
Z1 By N Square series Stub so you know that
00:30:05.609 --> 00:30:21.809
ABCD matrix its ABCD will turn out to be 1
this thing we have done earlier just refer
00:30:21.809 --> 00:30:32.490
there so the composite so RHS will be composite
of this this is the first one in this one
00:30:32.490 --> 00:30:49.749
if you do that you get 1 by 1 plus Square
then 1 J omega by N square Z1 minus Z2 J omega
00:30:49.749 --> 00:31:06.480
N square by Z2 1 minus gamma square Z1 by
Z2 now you see this is RHS this is LHS so
00:31:06.480 --> 00:31:17.990
see LHS is these RHS is these so on this two
becomes identical only when I choose that
00:31:17.990 --> 00:31:29.259
in value that N square is equal to 1 plus
Z2 by Z1 thats why kuroda has done that and
00:31:29.259 --> 00:31:36.809
in all these cases this N square 1 + Z2 by
Z N so with this you can now have all the
00:31:36.809 --> 00:31:46.609
permutations given and so if you want to design
any a implementable filter upto Richards transformation
00:31:46.609 --> 00:31:52.749
you come then you take the appropriate kuroda’s
identity because you see which one you need
00:31:52.749 --> 00:32:00.630
because if you want to if you want need if
you have a shunt capacitance but you want
00:32:00.630 --> 00:32:10.229
to convert with series 1 you can have this
just add this unit element and in series with
00:32:10.229 --> 00:32:16.619
that you can get it similarly if you have
this you can use this this also popular these
00:32:16.619 --> 00:32:25.980
2 these 2 sometime you use but generally inductance
and they are not generally in this fashion
00:32:25.980 --> 00:32:30.429
but some band pass etc they are there but
low pass high pass generally we have this
00:32:30.429 --> 00:32:36.840
type of thing so you can use these 2 identities
but if you can refer to here and always do
00:32:36.840 --> 00:32:45.519
this so we will see some implementation of
these in the next lecture that how we go about
00:32:45.519 --> 00:32:47.379
these transformations Thank You