WEBVTT
Kind: captions
Language: en
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Welcome to this thirteenth lecture of this
lecture series on design principles of RF
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filters and amplifiers you are discussing
RF amplifier and we have say in the last class
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we have seen that depending on the S parameter
values sometime we have unconditionally stable
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devices transistors sometimes we have potentially
unstable devices so we need to choose proper
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load impedance and source impedance so that
we can make the device conditionally stable
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that means a range of values of gamma L and
gamma S or ZLN and ZS needs to be defined
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for which the amplifier will be stable so
today we will see that and then we will see
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that is there any simplified way of finding
from the its S parameters whether the device
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is conditionally unconditionally stable or
potentially unstable so we can recall that
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in a loaded transistor amplifier that means
if I have a transistor with S parameter known
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an impedance level Z0 we will connect a source
similarly we will connect a load
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and we have seen that depending on impedance
level here there will be a reflection coefficient
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both sides that means towards the source that
is called gamma S there will be gamma S here
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there will be gamma in here similarly here
there will be depending on the mismatch in
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general there will be gamma L and there will
be gamma out also we have seen various gain
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definitions etc We have seen that transducer
power gain will give us the actual value of
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the power that is dissipated in the load by
power available from the source that ratio
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that means in the loaded condition that investor
the full system its power gain that is the
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useful criteria which will try to maximize
finally and achieve a amplifier design now
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last day we have filled that obviously our
gamma L and gamma S that means the corresponding
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ZL and ZS they are passive so they should
lie within the Smith chart unit circle immature
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and gamma L Gamma S they should be also within
the Smith chart with unit radius normalized
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with that unit radius and we have seen that
gamma in is given by S11 plus S1 to S21 gamma
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L by 1 minus S22 gamma L that we have already
seen and we demand that the device to be unconditionally
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or whenever potentially unstable also want
to be conditionally stable we require that
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gamma IN that means this should be less than
1 and also gamma out that is S22 plus S1 to
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S21 gamma S by 1 minus S11 gamma is this also
needs to be less than 1 otherwise we’ll
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have oscillations so now with which range
of gamma L and gamma S will enforce this condition
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that will see To do that we see a smith chart
there we will find out that actually you see
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the border region from here is that gamma
IN if it is less that 1 it is conditionally
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stable but if gamma In is greater than 1 it
is unstable similarly gamma out magnitude
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less than 1 it is stable conditionally stable
and come out greater than 1 it is unstable
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so these leads us to say that basically the
border region is this . that gamma IN is equal
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to 1 if we can locate this locus similarly
gamma out magnitude is equal to 1 then we
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can divide the whole smith chart into two
regions one will be stable region another
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is unstable region From the stable region
will choose our gamma L and Gamma S so these
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the boundary the boundary between stable and
unstable region in a smith chart is given
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a name it is called the stability circle Why
circle?
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Will just now see that it ultimately turns
down to be a circle this locus so this boundary
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that means locus of all these boundary region
the boundary point that becomes a circle that
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is called stability circle so in the you know
this smith chart we can consider these as
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a both as a reflection coefficient plot as
well as a corresponding impedance plot so
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in the reflection coefficient plane we can
say that stability circles are nothing but
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locus of gamma In equal to 1 or gamma out
equal to 1 in reflection coefficient plane
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of the smith chart
So that means we have two such locas’s so
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we will have two stability circle one we call
output stability circle that means which determines
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the value of gamma L that gives the boundary
that means for gamma IN is equal to one as
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you can see that gamma IN is basically a function
of gamma IN is basically a function of gamma
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L as well as parameters of the transistor
so this output stability circle is when gamma
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IN is equal to 1 that we call output stability
circle similarly gamma out is equal to one
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that we call in put stability circle because
these depends on the choice of source impedance
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or gamma reflection coefficient gamma S so
there are two stability circles Now let us
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see one by one . Let us first see the output
stability circle So output stability circle
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is given by gamma in magnitude is equal to
1 that implies we have S11 plus S12 S21 gamma
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L by 1 minus S22 gamma L is equal to one now
this we can now this equation we can manipulate
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remember this is values all r S parameters
and gamma L we are complex numbers also we
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can define this quantity gamma is the determinant
of the S matrix So gamma is nothing but S11
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is to 1 minus S21 S12 in general gamma is
also complex so these if we just manipulate
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these we if we solve for because you see this
is one equation where S11 S12 S21 S22 are
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known quantities gamma L is unknown quantities
so we can solve or gamma L and that gives
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that gamma L minus S22 minus delta S11 star
whole thing star or conjugate divided by S22
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Square delta square so you see this is simple
manipulation remembering the quantities are
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complex so gamma L can be retained as gamma
L is equal to this is a real number this is
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the magnitude of whole thing plus this complex
number and magnitude so in this complex plane
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this represents a circle because you see that
this is of the form I can write it as a form
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that gamma L minus some this is a complex
number so CL it is a complex number is equal
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to some real value so that I can say as RL
so these represents the circle that is why
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the same output stability circle so here CL
is the center of the circle RL is the radius
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of the circle so we can say that output stability
circle . it has it is circle is a as a center
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at CL is equal to S22 minus delta S11 start
whole thing star divided by S22 square minus
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delta square this is the center of the circle
you know that for the circle if I know the
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center and radius I can draw the circle knowing
the S parameter values or measuring them I
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can find what is the this circle this is the
radius of the circle . similarly I can have
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input stability circle which will be given
by gamma out is equal to 1 again gamma out
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you know is a function of gamma S so with
that we can find out gamma S if we properly
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rearrange then gamma S minus Cs is equal to
Rs this is the low cast of the circle of the
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input stability circle and what are this values
Cs turn out to be S11 minus delta S22 star
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star means conjugate
and RS is S12 s21 by S11 square minus delta
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square this is the center of the circle this
is the radius of the circle for memorizing
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you see that CL the S22 is replace by S 11
S11 is replaced by S22 here S22 is replace
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by S11 similarly this S22 is replace by S11
so equations are similar now what these circles
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do that means . if I
have a smith chart if like this then this
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stability circles they there are various choices
one thing can be stability circle is completely
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outside if this is stability circle that means
you see that they are not intercepting also
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I can have instead of these another stability
circle is like this it is completely enclosing
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the smith chart or the third choice is the
stability circle is intersecting with the
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smith chart so let us first see that intersection
part . so if we have this intersection one
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let us see this is the output stability circle
with somewhere centers CL and radius RL now
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what it says this is the output stability
circle so you see that these one has divided
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this whole smith chart into to region one
region is this portion and another region
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is the one that is inside the circle now since
this is a boundary this output stability circle
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means this is the locus of this point gamma
in is equal to 1 so I know that one of these
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either this region or this region will be
unconditional conditionally stable and another
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region is unstable now I will have to determine
which region is stable and which region is
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unstable so for that . let me draw these that
suppose this is my smith chart and this is
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my let us say output stability circle now
what I do there can be two cases that is suppose
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first to determine that which one is stable
whether this region is stable or this region
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is stable what I need to do in my thing if
you see that let me make a choice in my mind
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ZL is equal to Z0 that means I am choosing
the load impedance choosing this load impedance
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choosing this load impedance same as Z0 If
I do that immediately I can see that gamma
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I will become 0 so ZL is equal to Z0 makes
gamma I is equal to 0 and then I can see from
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this expression that if gamma n is equal to
0 gamma in magnitude becomes S11 magnitude
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so this is one of the choice as we have seen
in simple matching we do it instead of conjugate
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matching I can do this and in that case you
see gamma n becomes S11 Now there can be two
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possibilities this gamma S11 magnitude that
can be less than 1 or greater than 1 Let us
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say case 1 when it is S11 is less than 1 sorry
S11 is less than 1 now if S11 is less than
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1 immediately I can see that but let me write
here that gamma IN in this case becomes less
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than 1 which was my condition for stability
so that means and where this point lies you
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see gamma I is equal to 0 this point will
always lie at the center of the Smith Chart
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and I know that in this region at this point
gamma IN is less than 1 so this is a stable
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region immediately I got my answer that this
is the stable region of the Smith Chart So
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I can say this is a stable region and obviously
this one that means this portion of Smith
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chart is unstable so I will choose my gamma
L from the stable region then I will enforce
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that the input side is stable also now there
can be the second case that case 2 case 2
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what is case 2 that S11 value I do not have
any play with these because manufacturer has
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given is it can be greater than 1 if it is
greater than 1 and I have made this choice
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that Z1 is equal to Z0 so immediately see
that gamma IN again becomes S11 but this time
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it is greater than 1 so I know that this choice
that gamma L is equal to 0 this point lies
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at the center Let me again so this is for
this case Let me again draw these This is
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the Smith Chart and this is my output stability
circles and this is the point now in this
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case I see this is for case 2 this is the
case you see that gamma L is equal to 0 has
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led me to gamma N is greater than 1 so this
is an unstable region in this case so I immediately
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say that this zone is unstable and then I
know that this region is stable region so
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here sorry not this whole thing this part
inside because my gamma L should be passive
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so up to this point so then I need to choose
my gamma L only from this place if I choose
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it here I will get an instead of an amplifier
it will become an oscillator similarly I can
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do the whole exercise for input stability
circle . let this is my Smith Chart and let
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input stability circle is something like this
This is my input stability circle and that
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also has divided you can see again the thing
here and here that this portion and this portion
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so I need to now decide again I do the same
thing that I find out that let us take ZS
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is equal to Z0 that means here I choose Zs
is equal to Z0 immediately that makes my gamma
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S go to 0 and if gamma S go to 0 I know gamma
out is simply S22 magnitude so I got gamma
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out is equal to S22 magnitude and gamma s
is equal to 0 now I know gamma S this is my
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gamma S is equal to 0 point and this is S22
so again there can be case 1 that I have S22
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magnitude let less than 1 in that case I know
in case 1 know that gamma s equal to 0 so
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this is the stable region so I can immediately
say that this is my stable region stable and
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this portion unstable I should not find my
gamma S from here I should choose my gamma
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S anywhere from here so this is for case 1
so I can do the same exercise for case 2 this
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is my let us say smith chart and this is let
us say some other input stability circle and
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in case 2 I know that S11 can have greater
than 1 in that case my with this choice of
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Zs is equal to Z0 gamma S will become 0 and
gamma out that will be equal s22 that will
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be greater than 1 so I know that this point
of the smith chart as an unstable region so
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I will say that these becomes now my unstable
region and this is my stable region so I should
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choose my gamma S from here so this is for
intersection case as I said that . I can also
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have this that this is the smith chart and
any of the stability circle suppose input
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stability is this output circle is these output
stability circle input circle stability circle
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so that means there is no intro duct ion here
so I can easily find out that ok in this case
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if I locate that there are no intersection
means that device is unconditionally stable
00:26:45.710 --> 00:26:56.220
similarly I can also have the case that the
stability circle is completely like this ISC
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or C in that case also I see there are no
interaction so no problem I can have this
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thing carry on so by this so we can find out
that what are the ranges is gamma and gamma
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L for which we can have the conditionally
stable amplifier so we will choose accordingly
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to that but do we need to plot this circle
always that we will see in the next lecture
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that always that is not necessary sometimes
we can simplify just by some analytic expressions
00:27:36.590 --> 00:27:44.059
we can find out whether the device is unconditionally
stable if it is unconditionally we need not
00:27:44.059 --> 00:27:51.509
draw this stability circles but if it is conditionally
or potentially unstable then we need to draw
00:27:51.509 --> 00:27:57.580
this and locate this points look at these
ranges of gamma and gamma L for which I can
00:27:57.580 --> 00:28:05.499
make the amplifier stable and so I can use
the device even though it is potentially unstable
00:28:05.499 --> 00:28:12.710
but for my enforced condition of load impedance
and source impedance it will behave as a stable
00:28:12.710 --> 00:28:15.109
device stable amplifier thank you