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Hello everyone, and welcome to today's lecture
on Antenna Fundamentals. So, today we will
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talk about the basic characteristics of the
antennas.
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So, antennas are defined by its radiation
pattern. So, in general antenna radiation
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pattern is actually taken as a 3-D radiation
pattern though to make a life simple. We do
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plot 2-D pattern also, but let us say isotropic
antenna. Isotropic antenna is an antenna which
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radiates equally in all the directions whether
it is this direction or in this here, in the
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entire sphere it radiates equally in all the
direction. So, we define for that directivity
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is equal to 1 and we take 10 log of one which
is equal to 0 dB. I also want to tell, there
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are no isotropic antennas as such, but of
course, lot of research is going on to design
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an antenna which is known as quasi isotropic
antenna.
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Then there is a next thing which is Omni-directional
radiation pattern and what we have shown here
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is for a lambda by 2 dipole antenna. So, the
dipole antenna is kept over here as one can
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see the red shows maximum radiation and then
it is going towards lesser, lesser, lesser
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and least radiation. So, you can see that
if the dipole is like this here maximum radiation
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is all along the dipole antenna and if the
dipole if you see from here maximum radiation
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as we move up, up, up and if we see from the
tip, we only see the tip of the dipole or
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it is a very little radiation here and in
2-D pattern you can actually think of this
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whole thing as a 8 like this here and that
8 is getting rotated.
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Then here is an example of a direction antenna.
I have actually shown the radiation pattern
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of a microstrip antenna array which we have
designed at IIT, Bombay. So, you can see that
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there are number of elements are in this side
and number of elements are there and we are
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feeding these things. In fact, we have used
a series feed element over here and to feed
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elements in this direction we have actually
used slotted wave guide antenna here. So,
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for this particular antenna, we got a directivity
of 500 and if you take 10 log 500 which is
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about 27 dB. What you can see here is the
maximum radiation is in this direction and
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these are all the side lobes associated with
this particular antenna array.
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So, let say 2-D radiation pattern. This pattern,
we have already seen in the last lecture.
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But here we just want to define few additional
things now. So, just quickly recap, this is
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the main beam that is known as a major lobe,
all these are known as minor lobes. Half power
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beam width is defined as the angle over which
maximum radiation reduces to the half power
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point. For E field, it will be 1 then this
will be 1 by square root 2. After the main
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beam, there is a side lobe coming up, but
in between there is null. So, we have a null
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here and null here. So, the angular distance
between the two is defined by first null beam
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width. So, there is a simple relation between
the half power beam width and first null beam
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width.
In fact, beam width between first null is
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actually approximately given by 2.25 times
half power beam width. Well if you see many
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books actually write this even further approximation
which is two times half power beam width,
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but I have seen and checked with many antennas
and we have found that this relation is actually
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better than using a relation of two times
this one here. Then these side lobe levels,
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now side lobe level, we will see in time to
come that if we feed arrays with uniform amplitude,
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side lobe level is just about 13 dB or so.
However, for satellite and high power application,
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we want side lobe to be less than 20 dB. So,
just imagine a high power transmitter which
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is transmitting let us say 1 kilo watt power
in this direction. So, if it is 1 kilo watt
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and if the side lobe was just 10 dB, 10 dB
will be one-tenth of the power. So, 1 kilo
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watt is in this direction; that means our
100 watt power will be transmitted in these
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directions, so which is not desirable. So,
20 dB would mean 1 by 100. So, if it is 1
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kilo watt this is still 10 watt that also
may or may not be desirable.
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So, many a times high power application may
even demand 30 dB or 40 dB. So, 30 dB would
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imply if this is 1 kilo watt, 30 dB would
mean all of these are less than 1 watt power.
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So, lot of these restrictions are coming day
by day where people are going for high power
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and they would like a lower side lobe which
poses lot of challenges for the antenna designers.
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So, front to back ratio, generally preferred
is 20 dB, but 15 dB may be acceptable in some
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cases, but many a time the requirement could
be 30 dB, 40 dB front to back ratio.
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Then let us define directivity. So, we just
looked into the directivity three examples.
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So, directivity is 1, directivity is 1.6,
for directivity is 500. So, what is really
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directivity? How do we define? So, the basic
definition of directivity is that the directivity
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of the antenna is the ratio of radiation density
in the direction of maximum radiation. Please
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note, it is direction of maximum radiation.
So, for example, what is the radiation density
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in this, Divided by if it had radiated in
the sphere, so that is how we define directivity
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and that is given by maximum radiation divided
by average radiation intensity which is for
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let us say isotropic antenna and this one
can be because the power radiated if it is
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isotropic we say that power radiated divided
by 4 pi r square will be the power radiated
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in the spherical fashion 4 pi r square is
the area of the sphere.
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So, we are taking r is equal to 1. So, that
is what it simplifies here this 4 pi goes
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up here and this is the simple term which
is used to find the directivity, what is this
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term here? This is known as a beam solid angle
and this beam solid angle can be obtained
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from the radiation pattern and one can see
that one requires to do a double integration
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why double integration - because we have a
theta and phi. So, all these parameters let
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us say d phi will change from 0 to 2 pi, theta
will change from 0 to pi and that is how we
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need to integrate and what is F? F is nothing
but it can have both E theta and E phi which
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may vary in theta or phi direction.
So, you can naturally think that oh my god
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it is so complicated well in this course we
will try to make things simpler for you people
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we will try to use some of the approximate
formula to calculate the directivity with
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some reasonable accuracy. Now this solid angle
is approximated by this expression here if
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we think of this as a main beam. So, this
is the solid angle there. So, what is done
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is you approximate in the two orthogonal plane,
so half power beam width here and half power
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beam width perpendicular to that. So, this
solid angle is approximated as theta E times
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theta H. Now this expression do not use blindly
for all the cases we will just tell you when
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to use this expression and when to use modified
version of this expression.
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For example I have just taken an example of
infinitesimal dipole. So, for infinitesimal
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dipole the half power beam width is given
by pi by 2 and from where this pi by 2 comes?
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Because for infinitesimal dipole we will just
show here, so what we have here - the maximum
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variation it is coming down to 0 here. So,
from 1 let us say it is coming to 0 if we
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assume approximately this as a sin function
then we know that at sin 45 it will be 1 by
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square root 2. So, 45 plus 45 and minus 45
will give us 90 degree which is equal to pi
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by 2 and it is an Omni-directional pattern,
so in the h plane the beam width is 360 degree
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or 2 pi.
So, if I substitute these values in this expression
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over here, we get a number which is 1.3 whereas,
we know that for infinitesimal dipole antenna
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directivity is equal to 1.5. So, this number
predicts relatively smaller, but yet it is
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reasonably close to this value here and you
do not have to do any of these integrations
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also. We will tell you some simpler way to
also use this expression for larger antennas
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also.
So, for small antenna we can use the same
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expression which is 4 pi theta E theta H where
theta E theta H are in radiant. But now they
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are converted in degree.
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So, what you really do it is well you want
to convert from radian to this all you do
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it is from radiant conversion to degree is
you multiply by 180 divided by pi. So, over
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here then we have theta E theta H are in degrees
and that 4 pi after that 180 by pi accounted
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into becomes this particular number here.
Now this is for small antenna, but for larger
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antenna this is not a very good number in
fact, generally we use about same expression
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here we use about 32400 instead of 41000.
So, why we use this smaller number? The smaller
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number is used the reason for that is when
we are talking about the solid angle we are
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assuming that there is a only one main lobe;
however, in a given pattern there will be
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lot of side lobes are there. So, some power
is getting radiated in these directions. So,
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effectively lesser power will be radiated
in main beam.
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So, to account for this here this is an approximate
formula and if you use this formula invariably
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you will see that, you will get the directivity
expression within plus minus 1 dB. Now directivity
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is also related to the aperture area as we
saw in the last lecture if aperture is increased
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directivity will increase and again for a
frequency suppose let us say we take one frequency
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for that we have a certain aperture. So, for
example, let us say if I take a diameter of
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one meter for let us say circular dish antenna.
In fact, I have given an example here also
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we will come to that part also. So, from directivity
gain is defined as efficiency multiplied by
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directivity. So, let us say we want to find
the gain in dB of a parabolic reflector antenna
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at 15 gigahertz having diameter of 1 meter.
So, if the diameter is 1 meter, what will
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be the aperture area of this thing? Well aperture
area is given by pi r square; r is radius
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so in this case diameter is one meter. So,
r will be 50 centimeter.
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Now, that is A is now known what is lambda?
Well lambda has to be calculated from the
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frequency. So, here F is 15 gigahertz lambda
is equal to c by f where c is velocity of
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light. So, if you put that this particular
number comes out to be lambda equal to 2 centimeter.
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So, that is what you need to put over here.
So, lambda is 2 centimeter from there you
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can find the directivity; however, efficiency
of this parabolic reflector is only 0.6, so
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gain will reduce correspondingly. I just want
to mention here for a reflector antenna typically
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a maximum efficiency which can be obtained
is about 0.8, a poorly designed reflector
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antenna or purely manufacture reflector antenna
may give an efficiency of roughly 0.6.
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Now, I have just given an additional part
here the dish antenna diameter remains same
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what will be the gain at 36 gigahertz. So,
36 gigahertz the only thing which will changes
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lambda here. So, you can use that to find
out what is the directivity and gain of a
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reflector antenna. Instead of reflector antenna
I can give any other different problem also
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it can be an array let us say a square array
or let us say or rectangular array. So, of
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course, if the rectangular array has a dimension
of instead of diameter let us say the rectangular
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array has a dimension of 0.5 meter by 1 meter
then all you need to do it is effective aperture
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will become l multiplied by w and you can
do the calculation in the same fashion.
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So, polarization of the antenna we saw in
the very introductory lecture there are different
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types of polarization. So, here are more details
of the polarization. So, we have three main
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types linearly polarized, circularly polarized,
elliptically polarized. In fact, elliptical
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polarization is the most general form. So,
just think about if the major axis is equal
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to minor axis then actually this will become
circular polarization and if the minor axis
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goes towards 0 then this will reduce to the
straight line. So, that becomes linearly polarized.
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So, let us say that for the polarization to
be defined let us say any E component has
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theta component and phi component. So, E theta
caused this and then there is a phase difference
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between them.
Now, I will first talk about circular polarization.
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So, if E theta is equal to E phi and if the
phase difference between the 2 over here alpha
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is 90 degree. So, then we can say that the
magnitude of this or the vector rotation will
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be nothing but it will be in the circular
fashion and hence it is circularly polarized.
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So, here also it all depends upon whether
it is rotating in this fashion or whether
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it is rotating in the other fashion. So, it
is known as left hand circular polarization
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or right hand circular polarization. So, if
this case is not met then it can be linear
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or elliptical. So, in linear you can say that
if alpha is 0 or pi it will be linearly polarized
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in for all other cases it will be elliptically
polarized.
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Now as I mentioned elliptical polarization
is a general case. So, let us see how we define
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axial ratio of antenna. So, generally speaking
we want circular polarization for example,
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then for circular polarization axial ratio
should be 1, what is axial ratio? It is nothing
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but ratio of major axis divided by minor axis.
So, for circle major axis is equal to minor
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so axial ratio will be 1. For linear polarization
minor axis will have a 0, so 1 divided by
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0 will be infinity; anything in between will
lead us to elliptical polarization, but yet
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we define circular polarization approximate
circular polarization you can say that if
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the axial ratio this is the plot of axial
ratio versus frequency.
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So, you can see that this is axial ratio 1
2 3. So, if you draw a line over here. So,
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we can say that from this frequency up to
this frequency axial ratio is less than 3
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dB. So, generally speaking we define axial
ratio bandwidth for circularly polarized antenna
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as frequency range over which axial ratio
is less than 3 dB of course, in some application
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they may accept less than 6 dB also and in
some application they even want axial ratio
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to be less than 1 dB or 2 dB also. So, it
all depends upon application to application.
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Now, let us just look at the general input
impedance and VSWR of the antenna. So, input
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impedance of an antenna can be a complex quantity
which has a real component and imaginary component.
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The real component may actually consist of
the radiation resistance plus the losses associated
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with the antenna. Now this radiation resistance
is also kind of interesting thing in a sense
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it does not exist. It is only a mathematical
model. So, what is actually done? That power
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radiated from the antenna is considered as
power loss from circuit point of view. So,
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power radiated from the antenna can be approximately
written as let say i square R or v square
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by R. So, whatever is the power radiated divided
by the current or voltage that will give us
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the value of radiation resistance, this is
not really a real resistance which you think.
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So, when we talk about a 50 ohm impedance
of an antenna it is not that the resistor
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is 50 ohm; it is just a representation of
radiated power in terms of either voltage
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or current. So, we define efficiency as the
radiation resistance divided by the total
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resistance. So, what are the losses? So, losses
in the antenna can be dielectric losses or
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conductor losses. So, that is what gives rise
to the radiation efficiency then there is
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a next thing which is a reflection coefficient
and VSWR. So, reflection coefficient as we
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saw is defined by this particular term here.
So, if let us say Z0 is 50 ohm I just want
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to tell you that majority of the countries
in the world they have made a standard for
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microwave radiation and they actually use
impedance as 50 ohm. So, we will try to keep
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this 50 ohm in this particular course here,
but remember it can be different value, but
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if it is not specified assume 50 ohm.
So, now if antenna impedance is 50 ohm, 50
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minus 50 will be 0 we will get reflection
coefficient as 0 and if reflection coefficient
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is 0 power reflected will be 0 and we define
VSWR in terms of the reflection coefficient.
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So, this is just a practice problem for you
people. So, calculate reflection coefficient
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for I have given an example of let us say
only real impedance whereas, ZA can be complex
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in the next slide we will see what happens
if these are complex, but just to quickly
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tell you over here let us say we want to do
some quick calculation. So, if you put 100
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ohm over here for this particular case. So,
100 minus 50 will be 50; 100 plus 50 will
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be 150. So, this will be 50 divided by 150
will be 1 by 3 and if gamma is equal to 1
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by 3 we substitute the value that gives rise
to VSWR equal to 2. So, please calculate for
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other cases.
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Now, let us see what happens if the antenna
impedance is a complex number. So, here we
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have just taken an example of 20 plus j 30,
now there are two ways to solve this problem
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where we want to find out the reflection coefficient
and VSWR. So, one way is that we substitute
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in this particular expression gamma ZA minus
Z 0. So, Z A is 20 plus j 30, we put here
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we do the calculation simplify it. So, this
is the value which comes out to be reflection
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coefficient 0.56 angle this; and for VSWR
- we only need the magnitude we do not need
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the phase. So, we put the value over here
we get VSWR equal to 3.55. So, that is VSWR
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corresponding to the complex impedance.
Now, this is the one way to do it. The other
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way is to plot this input impedance on Smith
Chart. So, this is a Smith Chart here just
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to quickly tell you what is this Smith Chart
- the Smith Chart is a plot of input impedance
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or can be used for input admittance, but we
will keep it now for input impedance. So,
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what this plot is you can see here there is
a this central line here. This is actually
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a real access line. So, impedance here is
0 and it goes to infinity and generally Smith
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Chart is used for normalized impedance. So,
whatever is the characteristic impedance let
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us say in this case it is 50 ohm. So, we normalize
the value with respect to 50 ohms. So, you
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can see that Z normalized is given by this
here. So, what we need to do it is we need
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to locate this normalized value on the Smith
Chart. So, what we have here this is a real
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axis. So, on the real axis if you see the
Smith Chart it shows here 0 then 0.1, 0.2,
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0.4 at the center it will show exactly 1 and
then it will show 2, 5, 10, 20 and so on.
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So, on the real axis you locate the real value
0.4 which is this particular point over here.
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Now, then you will see there are so many other
circles are there. So, there are circles like
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this. So, these are known as constant resistance
circle, what this really means is that suppose
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I have chosen this 0.4 along this entire this
path here it will be 0.4 and if you just see
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here on this entire path the real impedance
will remain fixed then we have all these curves
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over here like this. These are truncated circles
and over here also these are truncated circle.
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Now this upper part is positive and the lower
part is negative imaginary part. So, for impedance
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if you are plotting impedance all the positive
values will correspond to the inductive impedance.
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All the negative values will correspond to
capacitive impedance.
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So, over here now what we have that these
are known as the constant reactance circle.
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So, for this entire this point curve here
the reactance value will remain fixed, but
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as you move from here to here to here only
the resistive value will change. So, our next
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part is to locate this impedance. So, 0.4
we located over here and then you move along
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this. Since it is plus, we need to move up.
So, we move up up up you stop at a point these
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are written here 0.1, 0.2 and, so you stop
at this point. So, this is the point which
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is the normalized impedance plot, now all
you do after that take this the central point
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draw this circle and wherever this circle
cuts this point now that is actually known
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as a VSWR. So, at this point here just read
whatever is the value, that value will give
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us 3.55.
Now, there is a another thing now you can
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also calculate the reflection coefficient
also there are two ways to do the calculation
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one is that actually below the Smith Chart
normally there is another horizontal line
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shown over here and you can actually see if
you just go down here this will show reflection
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coefficient as 0 and corresponding to this
it will show reflection coefficient as 1 and
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this is scaled from 0 to 1. So, if you just
draw this line here which is actually reflection
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coefficient 0 you draw just this line down
here and you can read directly the value and
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that will be 0.56. So, this is the one way.
Other way is, you measure this dimension and
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you measure this dimension using a scale take
the ratio and that will be 0.56 and you can
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measure the angle using a d which is there
in the normally in the compass box, you measure
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that angle that will give. So, this is Smith
Chart is nothing but graphical way of the
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representation of the Smith Chart and you
can calculate using this calculation.
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But I strongly encourage you people to practice
Smith Chart and just take some any other random
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number and do some practice for example, let
us say instead of 20 plus j 30 you can take
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as let us say 10 minus j 100. So, 10 normalized
will be 0.2 that will be somewhere here and
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this is if minus j 100 will be normalized
will be minus 2. So, 0.2 you go over here
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somewhere j 2. So, that will be the point
and then you can draw a circle and note down
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what is the VSWR correspondingly you can do
the calculation to verify your results.
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So, here is just an example I did mention
to you about microstrip antenna, just to show
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you. So, we had designed a microstrip antenna
at 5.8 gigahertz and we chose the feed point
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carefully so that we can get the impedance
match. So, here is the Smith Chart for this
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particular antenna if you see that this Smith
Chart is passing through the central point
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where impedance is exactly equal to 50 ohms.
So, there is an impedance match this is the
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reflection coefficient plot, which is a gamma
plot here. So, just as an example I can say
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that impedance plot which is normalized with
respect to 50 ohm and you can see there is
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a good matching and over here return loss
plot is there. Now generally, what we do we
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define bandwidth generally either for gamma
less than 10 dB or VSWR less than 2 just to
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tell you VSWR less than 2 corresponds to gamma
equal to 9.5 dB.
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So, this bandwidth here is about 85 mega hertz
you can see from here to here and this is
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at 5.8. So, you can take the ratio and that
will give us a bandwidth. You can see this
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bandwidth is relatively small. So, we will
see the techniques later on how to increase
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the bandwidth of the antenna.
So, just to summarize we looked into certain
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characteristics of the antennas. So, we looked
into 3-D pattern, then we looked into 2-D
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pattern, we looked into half power beam width,
first null beam width, side lobe levels and
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so on. We also looked at how to calculate
the directivity of the antenna, what is solid
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angle, what are theta E, theta H which are
two orthogonal component and how ban directivity
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has to be modified for larger array and then
we saw how to calculate the reflection coefficient
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and VSWR. We quickly looked into what is Smith
Chart, how to plot impedance on the Smith
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Chart and how to calculate reflection coefficient
and VSWR.
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And I strongly encourage that please practice
using Smith Chart and take some different
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00:29:28,919 --> 00:29:35,029
examples to do that and then we just showed
you one example of microstrip antenna how
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impedance matching can be done. Of course,
detailed of microstrip antenna will be covered
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much later in this particular course, but
we just saw that how impedance can be achieved
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and how we can find out the bandwidth of the
antenna from reflection coefficient plot or
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it can be obtained from VSWR plot.
So, thank you very much and in the next lecture,
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we will see some more antenna characteristic,
we will see how to find out that Friis transmission
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equation, the derivation and we will take
some more practical examples.
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Thank you, bye.