WEBVTT
Kind: captions
Language: en
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Welcome to this twelveth lecture in earlier
lecture we have seen various power definitions
00:00:31.050 --> 00:00:41.780
power gain definitions and we have related
the power various at various points in the
00:00:41.780 --> 00:00:51.250
microwave amplifier network now also we introduces
these three power gains now is the time to
00:00:51.250 --> 00:00:54.480
find the expressions for that power gain.
00:00:54.480 --> 00:01:07.490
. So G we know that G is equal to PL by Pin
if we put those expressions that we derived
00:01:07.490 --> 00:01:25.340
then we can write it as S21 square 1 minus
gamma L square by 1 minus 1 gamma in square
00:01:25.340 --> 00:01:41.880
1 minus S22 gamma n square so you see that
power gain it depends on load as we already
00:01:41.880 --> 00:01:49.290
said that it is independent of source that
is why you see the source reflection coefficient
00:01:49.290 --> 00:01:59.570
gamma is does not come here it is S parameter
of active device and various load conditions
00:01:59.570 --> 00:02:06.610
also from the network if we look that in just
a after Zs.
00:02:06.610 --> 00:02:13.020
If we look Z gamma in that will also independent
of Z2 that is why you get the expression also
00:02:13.020 --> 00:02:14.980
we get this.
00:02:14.980 --> 00:02:28.300
Similarly we can find out what is the available
power gain that we know is Pavn by pavs so
00:02:28.300 --> 00:02:40.230
it will be independent of ZL gamma L this
is again S21 square 1 minus gamma S square
00:02:40.230 --> 00:03:01.510
divided by 1 minus gamma out square into 1
minus gamma S square you see the similarity
00:03:01.510 --> 00:03:14.349
that gamma L is changes to gamma S gamma in
to gamma out S22 S11gamma L to this and from
00:03:14.349 --> 00:03:26.129
already derived ones we can now write the
transducer power gain that will be S21 square
00:03:26.129 --> 00:04:03.610
I said you see this transducer power gain
is dependent on both and source impedance
00:04:03.610 --> 00:04:13.560
that is why it depends on gamma S and gamma
L gamma In everything is included here now
00:04:13.560 --> 00:04:23.909
you see that sometimes this is this the maximum
gain possible but for this we require so GT
00:04:23.909 --> 00:04:30.659
is the maximum gain possible for a particular
gamma L gamma S which is conjugately match
00:04:30.659 --> 00:04:37.670
with the input and out impedances this is
a maximum that we can achieve and so please
00:04:37.670 --> 00:04:47.450
remember that this is for conjugate matching
input and output one sorry both but always
00:04:47.450 --> 00:04:58.700
we do not do such stringent ones because always
we need to know the loading conditions source
00:04:58.700 --> 00:05:07.719
impedance and load impedance to do this conjugate
matching sometime instead a easier one is
00:05:07.719 --> 00:05:19.909
done that is . just simple matching this we
can say simple matching what we to that we
00:05:19.909 --> 00:05:33.200
choose that Zs is
simple equal to Z0 you see Z0 is generally
00:05:33.200 --> 00:05:41.310
a real quantity so Zs and ZL we choose real
instead of that complex thing actual four
00:05:41.310 --> 00:05:48.300
com conjugate matching we require that Zs
should be chosen as Zin star ZL should be
00:05:48.300 --> 00:05:55.920
chosen as Z out star but simple matching is
this but if you look at the expression of
00:05:55.920 --> 00:06:03.720
gamma S and gamma L that we have done so if
I have Zs is equal to Z0 gamma s will become
00:06:03.720 --> 00:06:08.419
zero reflection coefficient there wont be
any reflection in the source there wont be
00:06:08.419 --> 00:06:14.700
any reflection in the load but that does not
ensure maximum power transfer that ensure
00:06:14.700 --> 00:06:22.300
less that maximum power transfer but sometimes
we do that reflection we just need to cut
00:06:22.300 --> 00:06:28.780
we are not so bothered about maximum power
transfer so in this case these implies gamma
00:06:28.780 --> 00:06:40.640
s is equal to gamma L is equal to 0 and then
what happens to these you can say GT simple
00:06:40.640 --> 00:06:52.190
matching this will be what is the expression
just now okay I have done so here if you see
00:06:52.190 --> 00:07:04.450
if put gamma S and gamma L 0 it becomes simply
S21 square so by simple matching what we can
00:07:04.450 --> 00:07:09.700
achieve whatever the transistor is giving
me whatever as we device it is giving me I
00:07:09.700 --> 00:07:20.459
can achieve that so let us see this first
graph yes . so you see that here I am not
00:07:20.459 --> 00:07:27.780
taking advantage of this graph so I am a good
RF engineer but this is sufficient that ok
00:07:27.780 --> 00:07:34.529
whatever device transistor is giving I want
transducer gain to to be that I can do that
00:07:34.529 --> 00:07:46.310
simply by simple matching another important
case is called in many practical transistors
00:07:46.310 --> 00:07:57.890
microwave transistor we have unilateral amplifier
. what is the meaning of obviously in an amplifier
00:07:57.890 --> 00:08:06.150
the S21 please remember that in a 2 port network
S21 means whatever am giving here how is coming
00:08:06.150 --> 00:08:15.089
here so this by this is S21 what is S12 that
if something is reflected how much is coming
00:08:15.089 --> 00:08:24.510
here in many of the many of the practical
amplifier this S2 value is very small practically
00:08:24.510 --> 00:08:34.029
I can all it zero sometimes it is not very
not exactly zero but very small compared to
00:08:34.029 --> 00:08:42.180
S21 it is quite small so if we consider it
to be zero so in that case let us look at
00:08:42.180 --> 00:08:50.000
our that this expression you see that from
signal flow graph I said that you can find
00:08:50.000 --> 00:08:58.510
out this gamma in is S11 plus this now here
S12 becomes zero then gamma even though you
00:08:58.510 --> 00:09:06.620
do not have a matching at the output gamma
in becomes practically S11 similarly if you
00:09:06.620 --> 00:09:16.440
write the expression for gamma out that will
be S22 so under this case we get that gamma
00:09:16.440 --> 00:09:30.180
in s11 and then the this transducer gain for
unilateral case will turn out to be S21 square
00:09:30.180 --> 00:09:48.510
1 minus gamma S square into 1 minus gamma
L square 1 minus S11 gamma S square 1 minus
00:09:48.510 --> 00:09:58.270
S22 gamma L square how it is different from
a transducer power gain for a non-unilateral
00:09:58.270 --> 00:10:08.330
that means in general one you see the GT expression
here there is the presence of this input reflection
00:10:08.330 --> 00:10:18.910
coefficient and output load reflection coefficient
this two sorry this input reflection coefficient
00:10:18.910 --> 00:10:25.720
now here that is absent that means input reflection
coefficient means I need to know what is loading
00:10:25.720 --> 00:10:32.190
here it is independent and it can be much
simpler design can be attempted to this so
00:10:32.190 --> 00:10:37.850
this is another special case that we will
see in the gain definition . but for unilateral
00:10:37.850 --> 00:10:47.780
amplifiers this simplification can be made
now let us generalize that we are in a position
00:10:47.780 --> 00:10:55.741
for simplify our whole microwave transistor
design I can say that instead if this design
00:10:55.741 --> 00:11:13.650
haaaa instead of please look at these two
figures where is the transducer power gain
00:11:13.650 --> 00:11:32.780
that three power gains not this hmm so
this was my actual circuit and this is the
00:11:32.780 --> 00:11:40.110
transducer power gain in the most generalized
sense now here you see that this is a total
00:11:40.110 --> 00:11:47.500
design that means I need to find out gamma
S gamma in etc . but now this whole thing
00:11:47.500 --> 00:11:59.940
I can attempt like this that I have a source
VS here I have a I have an input matching
00:11:59.940 --> 00:12:19.000
network so basically I have a Zs but with
a input matching here will have an input matching
00:12:19.000 --> 00:12:25.320
network this will be based on conjugate matching
and then I can say that with this input matching
00:12:25.320 --> 00:12:37.340
network that job of it is to bring the this
to an impedance level Z0 and then let us call
00:12:37.340 --> 00:12:55.480
this or that I will do later now then I have
the transistor S and then I have this there
00:12:55.480 --> 00:13:07.270
also will be output matching network and then
due to this output matching network instead
00:13:07.270 --> 00:13:17.600
of Zl I will say that I will terminate by
Z0 and so here I have that gamma S here I
00:13:17.600 --> 00:13:29.160
will have the gamma in here I will have that
gamma out and here I will have that gamma
00:13:29.160 --> 00:13:38.230
L so these two are equivalent things here
I has ZL Zs etc but that with input matching
00:13:38.230 --> 00:13:44.350
network and output matching network I can
do it and if we look here I will say that
00:13:44.350 --> 00:13:57.300
this part will give me a gain of Z0 this part
will give me the gain of source side again
00:13:57.300 --> 00:14:05.880
from this input matching network and this
I will get a GL load side input matching network
00:14:05.880 --> 00:14:14.430
and I will say my total GT is nothing but
GS, G0, GL.
00:14:14.430 --> 00:14:19.480
As you see that I have various produ get this
additional gains cts so if we group it together
00:14:19.480 --> 00:14:31.160
can I now say that what is GS ? GS is equal
to 1 minus you see this by this because source
00:14:31.160 --> 00:14:40.240
is coming here only so 1 minus gamma S square
source impedance, source matching they are
00:14:40.240 --> 00:15:05.730
coming here only GS I will have a GL is equal
to 1 minus gamma L square by 1 minus S22 gamma
00:15:05.730 --> 00:15:16.370
L square and the active part is G0 is s211
square.
00:15:16.370 --> 00:15:29.740
So you see that simple matching people they
we have already found that simple matching
00:15:29.740 --> 00:15:37.410
GT simple matching is S21 square but if I
do conjugate matching actually I have three
00:15:37.410 --> 00:15:43.830
parts so this is like simple matching but
i have two more degrees of freedom here one
00:15:43.830 --> 00:15:51.220
is the input source matching another is output
side matching so by this if I design properties
00:15:51.220 --> 00:15:57.860
two network I can get this additional gains
because here due to conjugate matching I will
00:15:57.860 --> 00:16:01.970
make maximum power transfer and by that I
can increase the gain.
00:16:01.970 --> 00:16:09.500
So basically you see when a transistor you
have already chosen ok obviously while choosing
00:16:09.500 --> 00:16:15.670
will choose a transistor which will suit out
purpose our impedance level etc and we will
00:16:15.670 --> 00:16:22.190
try to maximize this but once it is chosen
there is not much role for a designer to play
00:16:22.190 --> 00:16:30.520
but his role is basically to design here that
means if I want to have a high gain I need
00:16:30.520 --> 00:16:36.860
to play with these two basically so microwave
amplifier design is playing with this input
00:16:36.860 --> 00:16:43.089
matching network and output matching network
so that GS and GL can be made higher and higher
00:16:43.089 --> 00:16:49.339
obviously there is interconnection between
them independently this two cannot be done
00:16:49.339 --> 00:16:55.390
but we will do something we will see the techniques
by which this can be done.
00:16:55.390 --> 00:17:02.410
But there is another problem here that when
I am trying to make this gain maximize I should
00:17:02.410 --> 00:17:09.980
also understand that I can playing with an
active device so for certain loading condition
00:17:09.980 --> 00:17:14.010
because input matching network output matching
network design means I will have to choose
00:17:14.010 --> 00:17:18.350
proper leave my Zs and Zl levels.
00:17:18.350 --> 00:17:24.959
But to do that sometimes it the whole active
device become unstable what is unstability
00:17:24.959 --> 00:17:31.520
that for finite input it is giving infinite
output we know that condition is basically
00:17:31.520 --> 00:17:37.080
it can start oscillating etc., I do not want
amplifier to oscillate because that will start
00:17:37.080 --> 00:17:43.570
generating new more frequencies also its level
etc that will start going up so that we do
00:17:43.570 --> 00:17:49.870
not want from an amplifier so we want to do
a stability check for that now what is the
00:17:49.870 --> 00:17:57.760
so the next part before designing these finally
we will design these two that will be our
00:17:57.760 --> 00:18:04.420
base main aim but before that we want to do
a stability check for a particular load and
00:18:04.420 --> 00:18:10.720
source condition we want to see that what
is the stability of the amplifier because
00:18:10.720 --> 00:18:15.890
many times it happens in students project
that suppose you are asked to design an amplifier
00:18:15.890 --> 00:18:22.059
you see that finally you have loaded it such
that it becomes an oscillator so that means
00:18:22.059 --> 00:18:29.010
you have not given proper . attention to stability
so if we do that stability check so what is
00:18:29.010 --> 00:18:41.460
the stability criteria that basically Zin
if we see this that this input impedance Zin
00:18:41.460 --> 00:18:52.940
and also form this side I have output impedance
Z out now Zin if it has a negative real part
00:18:52.940 --> 00:19:01.780
that means instead of resistive if it has
a negative resistance also Z out as a negative
00:19:01.780 --> 00:19:12.390
real part obviously it is having an imaginary
part plus this is a reactive part but if either
00:19:12.390 --> 00:19:20.500
Zin or Z out or both they have negative real
part then I will start getting oscillation
00:19:20.500 --> 00:19:26.940
it is obvious because then whatever I am giving
suppose if Zin is more than it as a negative
00:19:26.940 --> 00:19:33.690
part whatever signal am given the signal will
start building up and then it will be starts
00:19:33.690 --> 00:19:38.950
oscillating similarly in the Z out whatever
reflection coming that will start picking
00:19:38.950 --> 00:19:41.650
up and the device will start oscillating.
00:19:41.650 --> 00:19:51.650
So the if Zin is having a negative real part
I can say basically that time the gamma in
00:19:51.650 --> 00:19:59.410
because what is Zin we know what is the relation
between gamma in and Zin gamma in is Zin minus
00:19:59.410 --> 00:20:07.850
Z0 by gamma by Zin plus Z0 so Zin if it as
a negative real part then I know that gamma
00:20:07.850 --> 00:20:13.980
in is greater than 1 this you know form your
smith chart knowledge also that in smith chart
00:20:13.980 --> 00:20:20.940
every point that represents either an impedance
or a corresponding reflecting coefficient
00:20:20.940 --> 00:20:30.100
now if Zin you know in the normal smith mart
that we use there the all we consider is passive
00:20:30.100 --> 00:20:36.710
impedances so they are the real part is never
negative real part of any impedance is never
00:20:36.710 --> 00:20:46.740
negative it is zero to infinity any value
it goes so what is the when Zin have a negative
00:20:46.740 --> 00:20:52.920
real part basically the point goes out of
our normal unit circle smith chart and that
00:20:52.920 --> 00:21:00.100
time gamma N becomes one similarly Z out is
here gamma out its magnitude that will be
00:21:00.100 --> 00:21:07.220
1 because in the smith chart that suppose
this is the point this is the magnitude of
00:21:07.220 --> 00:21:14.260
gamma now if the generally this radius is
one if gamma N greater than one means this
00:21:14.260 --> 00:21:22.130
point instead of here it is going here or
here it is going here like this so that means
00:21:22.130 --> 00:21:27.150
it is going out . of the passive smith chart
of unit circle.
00:21:27.150 --> 00:21:40.420
So what is the stability check we can check
this conditions that gamma in, if gamma in
00:21:40.420 --> 00:21:51.110
is greater than one magnitude of gamma in
is greater than one for certain or for any
00:21:51.110 --> 00:22:03.679
ZL and Zs combination then we say that device
is potentially unstable or I should say or
00:22:03.679 --> 00:22:09.179
gamma out is greater than one so gamma in
greater than one the gamma out is greater
00:22:09.179 --> 00:22:18.200
than one for it any choice of ZL any particular
combination of ZL, ZS then we say that device
00:22:18.200 --> 00:22:25.570
is potentially unstable so the whole stability
problem we break into these that I what is
00:22:25.570 --> 00:22:48.910
my this amplifier is unconditionally stable
IF and only if gamma in is less than one and
00:22:48.910 --> 00:23:01.940
gamma out both of this should satisfy gamma
out is less than one for all passive source
00:23:01.940 --> 00:23:13.770
Zs and ZL so amplifier is unconditionally
stable if for all passive because generally
00:23:13.770 --> 00:23:22.309
we are talking of passive source and load
impedances but this amplifier should be unconditionally
00:23:22.309 --> 00:23:30.360
stable now we have seen this gamma in and
gamma out values they depends on S parameters
00:23:30.360 --> 00:23:40.540
of the device and also load and source conditions
so we can find out this so we can find out
00:23:40.540 --> 00:23:48.160
some criteria or we can find on the smith
chart that whether this is taking place or
00:23:48.160 --> 00:24:13.740
not similarly amplifier is potentially unstable
if gamma in is greater than one or gamma out
00:24:13.740 --> 00:24:20.549
here not and because this was unconditionally
stable it is potential unstable if either
00:24:20.549 --> 00:24:32.830
these or this gamma out is greater than one
for some passive ZS ZL not for all even if
00:24:32.830 --> 00:24:40.580
or one then will say it is potential unstable
now we can make an amplifier conditionally
00:24:40.580 --> 00:24:47.730
stable that means though it is potentially
unstable will avoid that combination of Zs
00:24:47.730 --> 00:24:55.600
and ZL and then say that if I avoid this set
this particular choices of ZS ZL then I will
00:24:55.600 --> 00:25:11.070
make say that it is conditionally stable if
gamma in is less than one and again you see
00:25:11.070 --> 00:25:35.410
and gamma out is less than one for a certain
range of passive ZS and ZL so this is important
00:25:35.410 --> 00:25:43.030
because first we will see whether we can find
out whether it is unconditionally stable if
00:25:43.030 --> 00:25:48.620
the network is that unconditionally stable
that means what about the ZL ZS I choose still
00:25:48.620 --> 00:25:54.610
it is always stable then no problem I will
go ahead I will start concentrating my effort
00:25:54.610 --> 00:26:01.270
on design the input and output matching networks
so that we achieve the particular gain or
00:26:01.270 --> 00:26:07.010
maximum gain or other things but if it is
not if I say that no there is a potential
00:26:07.010 --> 00:26:13.190
unstability then first I will have to determine
what is the range of ZS and ZL that I can
00:26:13.190 --> 00:26:18.390
choose so that the amplifier becomes in the
stable region.
00:26:18.390 --> 00:26:39.740
So based on this we can say that . unconditional
stability implies that gamma in magnitude
00:26:39.740 --> 00:26:50.370
which we have already seen in earlier NPTEL
classes that it is S11 plus S12 s21 gamma
00:26:50.370 --> 00:27:07.460
L by 1 minus S22 gamma L that with should
be less than one and gamma out is equal to
00:27:07.460 --> 00:27:23.830
S22 plus S12 S21 gamma S by 1 minus S11 gamma
S that also to be less than one now if we
00:27:23.830 --> 00:27:35.930
have if the transistor is unilateral which
we already discussed that many practical transistors
00:27:35.930 --> 00:27:50.530
is unilateral then this condition become simplified
that gamma in becomes unilateral means S12
00:27:50.530 --> 00:28:07.130
is 0 so gamma 1 is S11 less than 1 and gamma
out S22 less than 1 so for a unilateral transistor
00:28:07.130 --> 00:28:12.630
I can always check because manufacturer always
gives me the S parameter or I can measure
00:28:12.630 --> 00:28:17.600
the S parameter by modern devices network
analyzer that we have already seen in earlier
00:28:17.600 --> 00:28:18.600
classes.
00:28:18.600 --> 00:28:25.921
So and if I find that ok the device is unilateral
it says S12 is equal to 0 this is means S12
00:28:25.921 --> 00:28:34.990
is 0 or approximately 0 and then I see that
S11 magnitude is less than 1 S22 magnitude
00:28:34.990 --> 00:28:41.970
is also less than 1 then I know that is input
reflection coefficient output reflection coefficient
00:28:41.970 --> 00:28:46.510
also will be less than 1 so I will be happy
I will go on designing the actual thing.
00:28:46.510 --> 00:28:56.919
If it is not then or if I can also fine that
S11 is greater than 1 or S22 is greater than
00:28:56.919 --> 00:29:03.150
1 then I will be worried and I will have to
do some more thing that some more thing is
00:29:03.150 --> 00:29:05.860
called the stability circle.
00:29:05.860 --> 00:29:16.740
I need to get a smith chart and out some stability
circles on that so that will be doing in the
00:29:16.740 --> 00:29:24.210
next class that will be seeing that how when
I have a potentially when the next class will
00:29:24.210 --> 00:29:44.110
be seeing that in potentially . unstable cases
how to make how to make conditionally stable
00:29:44.110 --> 00:29:56.410
amplifier so we need to demarcate some zone
of ZL ZS etc so that we wont touch that and
00:29:56.410 --> 00:30:05.840
will also demarcate the zone where we can
choose ZL Zs from those are stability circle
00:30:05.840 --> 00:30:14.370
drawing and determining the region where it
can have conditionally stable thing.
00:30:14.370 --> 00:30:21.740
So this is something like whatever you have
learnt in circuit classes that a any network
00:30:21.740 --> 00:30:28.059
that when it is become unstable you can find
that conditionally how to make it stable we
00:30:28.059 --> 00:30:29.950
will also see that in the next lecture.