WEBVTT
Kind: captions
Language: en
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Welcome to this lecture now in previous lecture
we have seen the amplifier design with
00:00:27.340 --> 00:00:34.500
conjugate matching we could not see an example
that how the input and output matching network
00:00:34.500 --> 00:00:45.530
are designed so an example we will see that
.00:40) so see this problem from pozar’s
00:00:45.530 --> 00:00:53.760
book. Again one of our reference in my edition
it is at page five seventy three a transistor
00:00:53.760 --> 00:01:01.640
at 4 gigahertz its S parameters are given
please remember that S parameter is a frequency
00:01:01.640 --> 00:01:07.990
dependent quantity. So manufacture specify
at various frequencies the S parameters so
00:01:07.990 --> 00:01:15.610
at 4 gigahertz we are testing it at other
frequencies it will also change. Also S parameters
00:01:15.610 --> 00:01:20.970
are bias dependent quantity so if we change
the DC biases. The S parameters values will
00:01:20.970 --> 00:01:30.320
also change so this is the problem S parameter
is given now you design aad for conjugate
00:01:30.320 --> 00:01:38.149
matching. Now first obviously we need to confirm
whether it is stable or not so we enforce
00:01:38.149 --> 00:01:45.860
rollest stability criteria . and find out
that K is greater than 1 also delta is less
00:01:45.860 --> 00:01:52.050
than 1 so transistor ank unconditionally stable
also it passes the Mu test that it should
00:01:52.050 --> 00:01:58.190
be . so conjugate match design possible we
know conjugate match means gamma S = gamma
00:01:58.190 --> 00:02:05.349
N star and gamma L = gamma L star sso we have
to do this simultaneously. And then we will
00:02:05.349 --> 00:02:10.979
get the gain of obtained gain maximum. This
are the gamma L, gamma out, gamma in gamma
00:02:10.979 --> 00:02:17.849
S that we have discussed also the corresponding
power criterias we have seen. So we see the
00:02:17.849 --> 00:02:24.680
design S1 again we are written the actual
values of S1 delta turns out to be this. Now
00:02:24.680 --> 00:02:32.739
as we have seen in the previous case that
we will have to find out the gamma L and gamma
00:02:32.739 --> 00:02:42.269
S. . This the solutions so with B1 gamma S
values so we will need to calculate B1 C1
00:02:42.269 --> 00:02:49.540
for finding gamma S . and B2 C2 for finding
gamma S and also seen rollets test passed.
00:02:49.540 --> 00:02:57.660
So we know that this will be satisfied to
gamma S and gamma L they can be octane. So
00:02:57.660 --> 00:03:06.859
you see we are doing that now see in the presentation
now that B1 turn out to be this, B2 is this
00:03:06.859 --> 00:03:15.109
C1 is this C2 is this. Remember B is are real
values C are complex quantities. So we can
00:03:15.109 --> 00:03:22.159
find the here we are calling gamma L with
a maximum because we are doing conjugate matching.
00:03:22.159 --> 00:03:31.409
So gamma L will be this .87 angle sixty one
degree and gamma S with a maximum because
00:03:31.409 --> 00:03:37.090
it is a conjugate matching that is why we
are giving a subscript M. . So we have got
00:03:37.090 --> 00:03:45.959
this values and from that we can easily find
out what is GTM that means the trans maximum
00:03:45.959 --> 00:03:51.549
possible transducer gain that this device
will give. So you see first part that means
00:03:51.549 --> 00:03:59.719
input part proper choice of gamma S that give
us a gain of 4. Roughly then S21 part that
00:03:59.719 --> 00:04:06.510
means a transistor gives a gain of device
itself alone gives as a gain of 6.7 roughly
00:04:06.510 --> 00:04:13.239
7 and this one 1.6. So you see that if we
did not do conjugate matching we did simple
00:04:13.239 --> 00:04:21.130
matching. We could not have extracted this
two parts so roughly 4.1 into 1.6 or in db
00:04:21.130 --> 00:04:28.590
scale you see this is converted to db. So
6 db and 2 db 8 db and RF hmm iip hmm hmm
00:04:28.590 --> 00:04:35.510
microwave designer can extract additionally
for the gain from the actual device whose
00:04:35.510 --> 00:04:44.930
gain is only 8.3 db. So this this are again
so what is that gamma in value you see this
00:04:44.930 --> 00:04:53.020
and gamma out is this. . Now let us do the
input matching that we need to do input matching
00:04:53.020 --> 00:05:04.979
because we have got gamma S value so ha you
see that where is the gamma S or see the
00:05:04.979 --> 00:05:19.500
previous one. Gamma S is 0.87 angle 123. So
in the smith chart let us plot this gamma
00:05:19.500 --> 00:05:26.700
S .87 it is inside you see this point and
123 roughly this. This is corresponding to
00:05:26.700 --> 00:05:33.259
Zs in previous figure. Zs is impedance seen
from the transistor by looking towards input
00:05:33.259 --> 00:05:39.479
matching section that means actually if I
have this transistor here so from here am
00:05:39.479 --> 00:05:50.139
looking Zs. I need the input matching network
to transfer this to Z0 so my job is from ZS
00:05:50.139 --> 00:05:56.470
I will have to go to Z0 which is the centre
of the Smith chart. So matching section is
00:05:56.470 --> 00:06:02.710
transforming Zs to source impedance Z0 how
we do it you know there are various ways that
00:06:02.710 --> 00:06:10.370
will follow the will make a after these section
our basically input matching network will
00:06:10.370 --> 00:06:17.979
be like this that I have this then I will
put a transmission line and then I will put
00:06:17.979 --> 00:06:26.840
a stub here either shunt or short hmm series
stub that will see various choices this are
00:06:26.840 --> 00:06:33.100
not unique but we will follow the particular
design already in the impedance matching section
00:06:33.100 --> 00:06:40.860
we have seen how to design such things. So
what will do will we have found Zs delta S
00:06:40.860 --> 00:06:51.389
. means this also con consider this as Zs.
Now since you convert that to Ys this is the
00:06:51.389 --> 00:06:58.390
Ys point now move towards source clockwise
till you hit this keep this 1+jb admittance
00:06:58.390 --> 00:07:05.112
circle. You are hitting it here so from here
so you know how much length of line of plane
00:07:05.112 --> 00:07:10.470
is . . because this is the same transmission
line. So you require this transmission line
00:07:10.470 --> 00:07:16.349
and then the length you can find from here.
So this give line length to be .12 to lambda
00:07:16.349 --> 00:07:22.479
to know here are those lambda scale so from
here you find from here to here upto this
00:07:22.479 --> 00:07:29.010
point I have point 1 to lambda and then here
you find out that what is the susceptance.
00:07:29.010 --> 00:07:34.050
Ith ith This is 1 + jb circle so there is
some susceptance that susceptance since it
00:07:34.050 --> 00:07:40.850
is admittance scale this upper portion is
– j so that is – j3.5 that you need to
00:07:40.850 --> 00:07:47.810
adjust by putting a stub . that we will do
we will. Here we have taken a open stub so
00:07:47.810 --> 00:07:54.560
for open means in the admittance scale this
is my open circuit. So I need go how much
00:07:54.560 --> 00:08:04.379
you see this is my minus J 3.5 so this if
I take it is .206 lambda so length is .206
00:08:04.379 --> 00:08:14.580
lambda. . So from there now we do the output
matching network you plot ZL it is here again
00:08:14.580 --> 00:08:21.270
convert it to YL then move towards load since
am moving towards load it is anticlockwise
00:08:21.270 --> 00:08:27.979
so move here you are here you are hitting
here susceptance again here is minus j3.5
00:08:27.979 --> 00:08:34.120
it contributed by open stub of length this.
. So this is the final design you see in the
00:08:34.120 --> 00:08:40.960
source side I have assumed all the transmission
line and stub they are impedance level is
00:08:40.960 --> 00:08:48.250
50 ohm same as our characteristic impedance
level of the transistor. So I have .12 lambda
00:08:48.250 --> 00:09:00.780
line length and then a stub 50 ohm stub similarly
here. A transmission line of length .206 lambda
00:09:00.780 --> 00:09:08.970
and stub is also 206 lambda so now from this
side the source will see a 50 ohm impedance.
00:09:08.970 --> 00:09:13.980
Similarly from this side the load will see
a 50 ohm impedance this is the conjugately
00:09:13.980 --> 00:09:21.231
matched amplifier ok. So this with this example
we conclude the conjugate matching iith then
00:09:21.231 --> 00:09:31.110
we do that always we do not do conjugate matching
. because conjugate matching is a you see
00:09:31.110 --> 00:09:36.030
that conjugate matching because all this things
are frequency dependent. So conjugate matching
00:09:36.030 --> 00:09:43.760
is possible for a particular frequency so
amplifier that means with conjugate matching
00:09:43.760 --> 00:09:50.600
design we achieve the maximum possible gain.
We have seen that I can extract 8 db extra
00:09:50.600 --> 00:09:56.590
gain etc in this case But that is over a narrow
band of frequency. So the whole amplifier
00:09:56.590 --> 00:10:03.220
design is narrow band sometimes we do not
need narrow band also that much gain may not
00:10:03.220 --> 00:10:11.580
be needed. So it is better to have a procedure
for specified gain. So design for specified
00:10:11.580 --> 00:10:22.750
gain
so let us have a procedure for this so again
00:10:22.750 --> 00:10:31.590
here. We generally now we are becoming more
practical and we are we have seen how to do
00:10:31.590 --> 00:10:38.060
bilateral design here we make an unilateral
approximation that generally all practical
00:10:38.060 --> 00:10:44.070
transistors that we will u we using unilateral
if you cannot make this unilateral assumption
00:10:44.070 --> 00:10:51.670
you know how to design here that you can inn
do. So when we design for specific gain which
00:10:51.670 --> 00:10:57.650
is less than the maximum gain possible, maximum
transistor gain possible. Definitely we are
00:10:57.650 --> 00:11:02.460
deliberately putting some mismatch in the
device we are not properly conjugate matching
00:11:02.460 --> 00:11:07.160
we are putting that mismatch that is why we
are getting less gain but we are satisfied
00:11:07.160 --> 00:11:15.080
with that because that must mismatched power
loss etc we can suffer sometimes if not then
00:11:15.080 --> 00:11:20.720
do the conjugate matching which is the best
design. So unilateral actually you see if
00:11:20.720 --> 00:11:34.050
I make unilateral thing actually a mine GT
in unilateral let us call that GTU. So actual
00:11:34.050 --> 00:11:42.941
device if it is not unilateral can I say that
I am committing an error of the order GT by
00:11:42.941 --> 00:11:53.820
GTU. So GTU is my actual thing GTU is my actual
thing people have shown that these as an bound
00:11:53.820 --> 00:12:03.490
that both lower bound and upper bound that
can be expressed by a parameter U, it is a
00:12:03.490 --> 00:12:13.060
unilateral case figure of merit. So this this
error or this ratio GT by GTU that means if
00:12:13.060 --> 00:12:20.790
I make this assumption am committing an error
or of them or my gain is . . by this that
00:12:20.790 --> 00:12:31.640
is have an upper bound and lower bound. Where
you is given by it depends on all S parameters
00:12:31.640 --> 00:13:02.140
of device, so this name of the parameter U
is called unilateral figure of merit and people
00:13:02.140 --> 00:13:18.080
have shown that usually this error it is with
in 0.1 to at the most 0.34 db db So I am not
00:13:18.080 --> 00:13:27.350
suffering in this if I make an unilateral
approximation then my error that I commit
00:13:27.350 --> 00:13:34.070
in getting the gain that is not much. So that
generally can be done but that simplifies
00:13:34.070 --> 00:13:43.600
the procedure, so let us see the procedure
for specified gain. . So specified gain so
00:13:43.600 --> 00:13:50.930
that means people will now tell me that you
get this much gain now as I already said that
00:13:50.930 --> 00:14:06.950
I have the whole gain that always in the GTU
case also I can divide as GSU G0 and GLU.
00:14:06.950 --> 00:14:14.280
So input matching section output matching
section now hence forward I leave this subscript
00:14:14.280 --> 00:14:24.500
U. So I can say that what is GS you know under
unilateral thing it is simplified that gamma
00:14:24.500 --> 00:14:32.800
L does not come into picture and it is simply
1- S11 delta S whole square so you see that
00:14:32.800 --> 00:14:41.370
GS I can choose just from gamma S or gamma
S I can choose by knowing what is my GS requirement
00:14:41.370 --> 00:14:48.840
that means input matching circuit how much
I want to have. Similarly GL can be retained
00:14:48.840 --> 00:15:01.410
as 1 – gamma L square by 1- S22 gamma L
square the moment I have this you see that
00:15:01.410 --> 00:15:07.030
output side I can design from the specification
of GL. So I have overall specification of
00:15:07.030 --> 00:15:13.970
this I know this value this is S21 square
of the transistor now this two in my hand.
00:15:13.970 --> 00:15:19.120
I can have any choice input matching and output
matching some gain some portion I take from
00:15:19.120 --> 00:15:25.530
here. Some portion I take from here and I
do that DB gain. So in the DB scale you know
00:15:25.530 --> 00:15:36.220
this is nothing but I can write it as GTU
in DB is GS or let me under all are under
00:15:36.220 --> 00:15:48.740
this GS db +G0 + GL DB. So there you can see
that I can choose that ok. Suppose if it is
00:15:48.740 --> 00:15:55.140
11 db required suppose this is 8 db so 3db
now it is upto to distribute either I can
00:15:55.140 --> 00:16:09.060
2 db here 1 db here or 1 db here 2 db here.
Now out of this you see this GS value GS is
00:16:09.060 --> 00:16:27.250
maximum if I can choose gamma S = S11 star
also this GL will be maximum if I can make
00:16:27.250 --> 00:16:38.360
gamma L = S22 star if I can do that then immediately
there will be maximum this thing just from
00:16:38.360 --> 00:16:49.730
here you can see and I can write that value
that what is GSM that will be GSM that is
00:16:49.730 --> 00:16:59.340
1 by 1 – S11 square. So maximum I can do
depends on the S11 parameter. Similarly from
00:16:59.340 --> 00:17:10.809
the output matching section GLM I can see
this 1 by 1 – S22 square so now what you
00:17:10.809 --> 00:17:26.850
do to facilitate the design we define a normalized
gain function normalized gain function . so
00:17:26.850 --> 00:17:37.330
I have a 2 such gain functions GS small gs
and small gl. What is GS which is my specified
00:17:37.330 --> 00:17:50.250
by the maximum that I can get similarly what
is GL it is GL by GLM and what is this if
00:17:50.250 --> 00:17:57.559
I write the expression all the expression
are there. You simply write it 1-delta S square
00:17:57.559 --> 00:18:10.750
by 1- S11 delta S this is my gs and maximum
is 1 – S11 square this was denominator so
00:18:10.750 --> 00:18:23.320
it as gone here. Similarly here I can write
1 – gamma L square by 1- S22 gamma L square
00:18:23.320 --> 00:18:36.690
1- S22 square very elegant and it is obvious
that 0 this is the benefit of normalization
00:18:36.690 --> 00:18:47.350
that this normalize gain function are line
in between 0 and 1 and now you see this one
00:18:47.350 --> 00:18:54.620
I can just the manipulate as I have done incase
of the conjugate matching what we did we have
00:18:54.620 --> 00:19:03.980
retained GS. As a function of GL and GL as
a function of GS and then 2 equations to unknown
00:19:03.980 --> 00:19:09.950
we did that but here already input and output
are uncouple. So form 1 equation suppose from
00:19:09.950 --> 00:19:16.740
this equation I can find out what is solution
for gamma S from this equation I can find
00:19:16.740 --> 00:19:22.799
out what is solution for gamma L I can choose
accordingly. So we will do that accordingly
00:19:22.799 --> 00:19:31.629
we write this GS equation now forget this
GS is this from there we find out the solution
00:19:31.629 --> 00:19:38.519
for gamma S. One equation one unknown I can
always do that . so GS I write as 1 minus
00:19:38.519 --> 00:19:50.620
again I am writing 1- gamma S square by 1-
S11 gamma S square into 1 – S11 square.
00:19:50.620 --> 00:20:39.370
So solve for gamma s if I do that again I
get something like this so you see again this
00:20:39.370 --> 00:20:48.360
is complex number this is a complex number
this is a real number small gs is a real number.
00:20:48.360 --> 00:20:58.730
Now what i this in the smith chat if you plot
this or this locus can I say this same as
00:20:58.730 --> 00:21:04.980
before this is the equation of a or locus
of is this locus will be a circle, this is
00:21:04.980 --> 00:21:12.720
center, this is the radius, so we call that
now what is this ? This circle is a specified
00:21:12.720 --> 00:21:27.940
value or a constant value so we call this
as constant gain input circle. Because here
00:21:27.940 --> 00:21:34.649
we are choosing the input side so constant
gain input circle or constant gain circle
00:21:34.649 --> 00:21:52.950
for input section. So what will be my Cs,
Cs will be Gs S11 star by 1 – 1- Gs S11
00:21:52.950 --> 00:22:09.400
square and what will be my radius? Radius
will be root over 1 – Gs 1 – S11 square
00:22:09.400 --> 00:22:22.749
by 1 – 1- Gs S11 square Gs is specified
S11 value I know from the S parameter so I
00:22:22.749 --> 00:22:29.530
can find out this and I can plot that and
the smith chart that will be a circle. Similarly
00:22:29.530 --> 00:22:51.450
I can do for the other thing that the constant
constant gain output circle .. So here we
00:22:51.450 --> 00:23:00.320
have solve for Gs there will be from this
normalize gain function you solve for gamma
00:23:00.320 --> 00:23:17.539
L that locus again a circle and there the
CL the center will be GL S 22 start by 1 – 1-
00:23:17.539 --> 00:23:36.190
GL S22 square and radius is root over 1 – GL
1-S22 square divided by 1-1- GL S22 Square.
00:23:36.190 --> 00:23:49.480
You
see so from these we will see an example how
00:23:49.480 --> 00:23:58.639
to choose but here let us see this two that
this constant gain input circle and constant
00:23:58.639 --> 00:24:08.429
gain output circle. There are some points
that needs to be underscored that we can see
00:24:08.429 --> 00:24:17.029
that various values of gains. I can have . .
because as I said suppose I decide input
00:24:17.029 --> 00:24:31.529
side I am not sure whether to take 1 db or
2 db or 0 db or -1 db which gain I will take.
00:24:31.529 --> 00:24:42.590
But one thing is true that centers of each
family
00:24:42.590 --> 00:25:03.879
lie along the straight line joining center
of smith chart to either S11 star or S22 star
00:25:03.879 --> 00:25:13.890
whatever the cas S22 star. You see the center
is this is all some real number so Cs is along
00:25:13.890 --> 00:25:22.110
S11 star in the smith chart that is a complex
number it is along S11 star or CL L is along
00:25:22.110 --> 00:25:42.559
S22 star. Then another point is when Gs = 1
or Gl =1 what happens you see if Cs Gs
00:25:42.559 --> 00:26:01.009
= 1 then it becomes simply Cs from S11 star
and this center becomes S22 star. So then
00:26:01.009 --> 00:26:18.769
center simply S11 star or S22 star as the
case may be also it can be shown that 0 db
00:26:18.769 --> 00:26:37.669
gain circles pass thru center of smith chart
so from there you can always draw the circles
00:26:37.669 --> 00:26:45.940
plotted. And then you can choose which gamma
S and gamma L value so that you get the desired
00:26:45.940 --> 00:26:51.590
gain. Now this choice is not unique, there
is no unique design this is your a but one
00:26:51.590 --> 00:27:00.100
thing you try to choose gamma S and gamma
L as near to the smith chart center as possible
00:27:00.100 --> 00:27:06.419
because that will reduce mismatch though you
are deliberately giving some mismatch you
00:27:06.419 --> 00:27:12.830
cannot choose the center value but if you
choose it keeping your gain requirement if
00:27:12.830 --> 00:27:18.649
you choose it as near to center as possible
that will be better. Amplifier design for
00:27:18.649 --> 00:27:27.159
specified gain so here you see unilateral
case this are all this thing again unilateral
00:27:27.159 --> 00:27:34.279
figure of merit .:27:28) so pozers problem
page 625design an amplifier to have a gain
00:27:34.279 --> 00:27:43.100
of 11db at 4 gigahertz. The transistor is
given with this the first check the unconditional
00:27:43.100 --> 00:27:50.830
stability S11 is less than 1 S22 less than
1 you see this is unilateral already S12 is
00:27:50.830 --> 00:28:01.029
0 so maximum value will be obtained here.
So find out those values . and you see the
00:28:01.029 --> 00:28:08.679
I have written everything so GTU GO from the
input section here instead of S am calling
00:28:08.679 --> 00:28:17.059
G1. So input section can give you maximum
3.6db the output section can give you 1.94
00:28:17.059 --> 00:28:27.409
db and the transistor itself gives you a gain
of 8 db. So maximum you can have 3.6 + 8 +
00:28:27.409 --> 00:28:35.600
13.5 db but it is required 11 so you can cut
down here or here as per your thing. So you
00:28:35.600 --> 00:28:42.830
draw constant gain circles determine so let
us say that we have determine here suppose
00:28:42.830 --> 00:28:52.419
this is the 3 db circle in the input side
the red one is 3 db circle green one blue
00:28:52.419 --> 00:28:59.960
one is 3 db circle. Now where I will choose
you see that a prudent choice will be as near
00:28:59.960 --> 00:29:05.440
to as possible so if I decide I will take
2 db from input side I will choose the point
00:29:05.440 --> 00:29:13.690
here that is a good choice I can choose also
here here but if I choose it here that will
00:29:13.690 --> 00:29:21.919
be more prudent. . So here you see that gain
is this normalize gain factor you see distance
00:29:21.919 --> 00:29:28.200
of circle radius of circle. So by that we
are plotted constant gain circle . so for
00:29:28.200 --> 00:29:36.289
3 db circle this are the values given. Similarly
for the 2 db this are the values given similarly
00:29:36.289 --> 00:29:47.970
for the output side you see we have chosen
this two that for 2 db circle like this . then
00:29:47.970 --> 00:29:54.470
choosing we have decided that already 8 db
the device was giving 3 db we decided from
00:29:54.470 --> 00:30:00.919
the input we will take 2 db and output we
will take 1 db. So choose this you see I have
00:30:00.919 --> 00:30:06.379
chosen the nearest point for this so that
gives the value if I read this value it is
00:30:06.379 --> 00:30:15.390
.32 one twenty degree and for the load we
have chosen .22 seventy degree. . Now gain
00:30:15.390 --> 00:30:24.789
you can find this . you can design the matching
networks because the moment you choose gamma
00:30:24.789 --> 00:30:32.440
S is this you can find out locate that point
gamma S then gone back to YS. Then C where
00:30:32.440 --> 00:30:40.010
the cutting 1 + Zs circle from there you can
design the stub so here I have seen that .18C
00:30:40.010 --> 00:30:47.980
.18 lambda you will have to have the transmission
line then you have to cancel this stub. . So
00:30:47.980 --> 00:31:01.799
that similarly gamma S so here I have given
all the values and by that again I do the
00:31:01.799 --> 00:31:07.499
input matching and output matching. So this
is the final circuit you see I have used a
00:31:07.499 --> 00:31:13.450
transmission line then a stub of this much
length then here also a transmission line
00:31:13.450 --> 00:31:21.070
and stub this length so this constant gain
thing that gives you 50 ohm here 50 ohm here
00:31:21.070 --> 00:31:26.740
though actually the source and load impedance
where different but we are matching now it
00:31:26.740 --> 00:31:34.820
appears as 50 ohm here 50 ohm here. So this
is the total design for given 11 db thing.
00:31:34.820 --> 00:31:42.129
I hope now you understand how to design for
constant gain circle two more things we need
00:31:42.129 --> 00:31:51.529
to see that one is the sometimes we need to
have instead of gain another very important
00:31:51.529 --> 00:31:56.169
parameters sometimes particularly for L and
S that is the noise figure so we will have
00:31:56.169 --> 00:32:04.190
to make a particular noise figure or minimize
the noise figure gain may be anything moderate
00:32:04.190 --> 00:32:09.950
gain so that design we will see that design
also. We will see broad band design also we
00:32:09.950 --> 00:32:17.529
will see that sometimes in a power amplifier
the non linearity things comes stability we
00:32:17.529 --> 00:32:25.230
are enforcing everywhere in a every amplifier
but if I give very high input then non linearity
00:32:25.230 --> 00:32:31.879
comes so how to design under that non linearity
or keep that non linearity away. How to design
00:32:31.879 --> 00:32:39.299
the amplifier if my amplifier as non linearity
then sometimes I will give rise to additional
00:32:39.299 --> 00:32:47.389
spurious frequencies which is non desirable.
So how to make power amplifier design if we
00:32:47.389 --> 00:32:52.070
do that this whole RF amplifier design part
will be completed. Thank you