Antennas

Prof. Girish Kumar

Department of Electrical Engineering
Indian Institute of Technology, Bombay

Module – 12
Lecture – 58
Reflector Antennas – III

Hello, and welcome to today’s lecture on Reflector antennas. In fact, today’s lecture is in
continuation with the reflector antennas which we were discussing in the last couple of
lectures. So, we have started with the plane planar reflector antenna. Basically which was a
flat reflector, and in the flat reflectors we talked about two different categories: one was
planar reflector and then other one was corner reflector.

So, for planar reflector we actually looked into if the dipole antenna is kept in parallel with
the ground plane or in perpendicular to the ground plane. And we saw what is the effect on
the gain and the radiation pattern. After that we looked at different corner reflector antennas
and we saw that 30 degree, 45 degree, 60 degree, 90 degree reflector antennas will form
different number of images. And number of images is nothing but equal to 360/( -1); where
 is the corner reflector angle.

So, we have seen that for 90 degree corner reflector there were 3 images and for 30 degree
corner reflector there were 11 images. And basically if numbers of images are more than that
would really imply that the gain of the antenna is large. A typically a corner reflector antenna
may give a gain of the order of 10 to 12 dB. But today we will talk about parabolic reflector
antenna which can give gain of 20 dB, 30 dB, 40 dB, 50 dB, 60 dB and even 70 dB. And just
to look into the number so 50 dB corresponds to 100000, and 60 dB corresponds to 1 million.
So, these are the very large values of the gain and of course, it requires very large reflector
antenna.

So, let us start with the reflector antenna. And before we start with the reflector antenna
please remember one thing reflector antenna majority of the time uses a parabolic dish
antenna, and parabola has a property that if we have a parabola and something which is
coming from the infinity then the waves after reflecting from the surface will focus at a focal
point. Or if we put a feed at the focal point and if it is transmitting then after reflection from
the surface it will go and in parallel and it can cover a very large distance depending upon the
fresh transmission equation which we discussed in the earlier few lecture.

(Refer Slide Time: 02:54)

So, today now let us see the parabolic reflector antenna. So this is; what is a parabolic
reflector antenna which is shown over here, and we can actually see that there is a feed which
is put at the focal point. And the property of the parabola reflector is that if the feed point if
you looked at the focal point then the distance from here let us say; to this one up to the
surface and reflect back here. So, this distance remains constant for any angle of θ. For
example, if we say this is O, this is P and this is Q. So, OP+PQ is constant, and that would be
OP. If you take this point here let us say this is P’ and if this will be Q’ then OP’ + P’Q’ will
be same as OP+PQ. And then we can generalise that from here to here that distance is f which
is the focal point. So, this plus this will equal to OP+PQ.

So, for a parabola we can actually write a condition that OP+PQ is constant and that is equal
to 2f. So now, we can actually write this in a slightly different form. So, OP is nothing but let
us say we define this distance as r’ and that will be OP is r’. And what is PQ? PQ is over here.
So, if this angle is θ then if you draw the line here this angle will be also θ. So, if this is r’
then r’ multiplied by cos θ will be equal to PQ. So, that is PQ is equal to r’cos θ’.

So, we can combine this particular equation as OP is r’ and PQ is r’cosθ, so we can take r’
outside and the right hand side will be 2f. And this can be simplified as r’ = 2f this whole
thing comes over here. And from here the simplification comes in the form of; so we know
that let us say cos 2θ = 2cos2θ - 1. So, this is what it is; so this is cos θ. So, that will be 2 cos2
θ’/2 – 1, minus 1 and 1 will cancel. And now 2 cos 2θ’ will be in the denominator so that will
become sec2 θ/2 in the numerator and the two term which came here will get cancelled.

Now, this is the equation of a parabola in the polar coordinates. In fact, many a times people
are familiar with more like y2 = 4ax form that is a Cartesian coordinate, but this is a polar
coordinate. So, here a just few other things I want to highlight. So, the focal length here is f
here, and this angle which is maximum angle will be θ0 that is the angle made by the outer
edge of the parabola.

Now that if you look at the parabola even though it looks like a line, but if you look from the
complete 3D point of view this whole thing will look like a circle. So, we can see that this is
nothing but a circle and whose diameter is given by d. So, we can actually say that from here
to here distance will be d/2.

So if you look at the aperture; the aperture of the parabolic reflector will look like a circular
aperture. And for that circular aperture we can also find out what will be the gain. So, what is

4 πA
the gain for a circular aperture, we know that the gain is defined as η . So, where A is
❑2
area; so in this case area will be A is nothing but πr2 that will be equal to πd2/4. So, from that
we can find out what is the gain value.

But right now let us just look in to this equation one more time. So, we can expand this
equation in the Cartesian coordinate. So, how do we define r’+ r’cosθ’ that is the left side; so
that is defined that r’ is nothing but √ x2 + y 2 + z 2 . And r’cos θ’ is nothing but in this
particular thing which is z’ that comes over here and 2f is as before in the right hand side.

Few more things I just want to highlight here. So, here z is measured in this direction, y is
measured in this direction which is a y axis, and perpendicular to this will be x axis. So x is
here, y is here and this is z axis. So, now if you take this z’ term on this side and square both
the side. So, if you square then square root will go away we will be left with x 2+ y 2 , but
now z 2 gets cancelled because this term which is going in this side so this will be now like
(a-b)2 which is given by a2 - 2ab + b2. So, b here is z’, so that will be z 2 on this side; that z2
gets cancelled over here. And this term over here will be one term will be 4f2 and then minus
of 4fz’ which is simplified over here.
So now, one can see that this equation is given by x 2 + y2 = 4f (f - z’). And what is the
limiting condition of this? Limiting condition will be when x’ comes at the end, so x’ 2 + y’2
≤ (d/2)2. So, this point here you can say if it is somewhere here then; that means we have a
at this particular points x’ will be close to you can say 0 and y dash will be also close to 0 at
this particular point. But when we look at the extreme end over here we can say that this will
be the equation of the circle which is given by x2 + y2 = (d/2)2.

Now, this equation can be further simplified and will see what happens in the next slide.

(Refer Slide Time: 09:44)

So, here now we are few quantities to be defined, this is a tan -1 if we take this side tan θ0 is
given by (d/2)/z0. So, let us go back what it is? So tan θ0, θ0 is this one here. So, tan θ0 will be
this distance divided by this distance here. Now this distance we know is nothing but half of
the diameter. So, that will be d/2, and distance from here to here we have defined that as z 0
over here. So, that will be this distance z0.

And now we can further simplify this whole thing. So, tan θ, you can see that d/2 and then z0
these terms you will need to solve this particular thing, use the equations given in this
particular slide we know that what is z’; z’ can be simplified in terms of r and cos θ and then
simplify you can use this particular equation to find out the limiting case. You can see that
over here we can put the limiting case. So, if we put the limiting case here this will be let us
say at the edge x2 + y2 = (d/2)2. So, that will be this term over here. And this will be then 4f 2 -
4fz’ and z’ will be equal to z0 when we are looking at this particular point.
So, by using this equation finding the value of z0 from here substituting the values over here
and doing little bit simplification we get this particular equation over here. And that is the
next step we are basically what has done is simply this has been divided by (f/d) 2. If you
divide by everything (f/d)2 will be left with -1/16 here and if you divide here (f/d) 2 then x will
become square, and if you divide this (f/d)2 you can see that one term will go away this is
what we will get.

So, what we can see here there is a relation between θ0 and f/d. So, if f/d is given for a
parabolic reflector we can find the value of θ0. And this equation can be further simplified
and we can right this whole thing as tan θ0 over here and that gets simplify to this term. And
this term what we can see here that if θ0 is known then we can find out what is the value of f
for a given d, or we can bring this d this side. So, that will be f/d will be equal to 1/4cotθ0/2.

In fact this is the very very important thing, and one should really look at these values
because this will help us in designing a reflector antenna. It is very very important to choose a
proper value of f/d, as we will see later on. So, let us just see I have just done some
calculation here.

So, f/d; so d comes on this side, so if f/d is 0.4 you can either use this equation and we have
given the different values of f/d 0.4, 0.5. And then correspondingly the value of θ0 has been
calculated so that comes out to be 64, 53 and so on. Now, the question is what value of f/d we
should choose? Or alternatively what value of θ0 we should choose? So I will just give a
general recommendation. Generally f/d equal to 0.5 is chosen for majority of the practical
parabolic reflector which are prime focus reflector. Later on we will also talk about
Cassegrain reflector.

And we will see that for majority of the time for Cassegrain reflector f/d is tending closer to
value of 1. So let us just see here, let us go back to the last slide and we will see that; what is
f/d 0.5 and 53.1 really imply. So, what we are looking at is this angle of roughly 53 degree
and that will give the value of f/d. That means, now we should know what should be the
diameter and then correspondingly we should choose f/d.

So, this actually the whole thing starts as a design concept. And if you recall in the very
beginning when we are talking about the efficiency and we were talking about the Friis
transition equation we had also talked about directivity related with the aperture. So, let us
just look very quickly in to a design of the reflector antenna.
Now, majority of the time actually speaking the problem starts with the design specification.
So, the design specification could be; let us say design of parabolic reflector antenna for a
gain of say 40 dB at frequency of say 4 gigahertz. So, that would be the only thing which
would be given to a designer, nobody will give you what is the f/d ratio or what you should
do where you should feed point; all it is specified majority if the time is well gain is given
and you design the whole reflector antenna.

So, this is where the starting point is. So, let us say now we know G = 4πA/2, and A which is
area and the area what we have to take we have to only take the area of the surface here, not
this depth does not come into picture at all as far as the gain equation is concerned. But it
does come indirectly in the form of the efficiency. So, let us just relook into it. So, the gain is

4 πA πd 2 .
nothing but
❑2
η , area is πd2/4, so gain is now equal to ( )
❑
η

Now we know the gain we know the frequency, and now we can calculate the value of d.
Now of course, we need to know the efficiency, and as we will see a in the next few slides
typical efficiency of a reflector antenna can be about 0.7 to 0.8. However, they are some
papers which theoretically do claim that you can get an efficiency of close to 100 percent, but
those are really very very tedious thing. Right now we will focus on the prime focus reflector
where we can get typically efficiency of 70 to 80 percent and not more than that.

2
Now this equation which we just said G= πd η . So, that gives us for a given value of
( )
❑
gain and frequency we can find out what is the value of the d. It does not talk about f/d. So,
how do we calculate and how do we know what should be the value of f/d? So, this is where I
gave the general guideline that f/d typically can be taken as 0.54 prime focus. In fact, the
configuration which I am showing you is a prime focus.

So, now what really happens? So, suppose if f/d is small, so what will happen? f/d small that
means focal point will be closer. So, the focal point is closer than the reflector will be more of
this shape which will have a larger bend, but if f/d is increasing; that means this focal point is
changing in that particular case this reflector will keep opening up. So for a larger f/d the
reflector will look more like a flatter thing, more like a flat plate; and if you bend like this it
looks more like a flat bowl. So, you can think about soup bowl, so soup bowl will have a f/d
is close to even 0.5. And if you look I can soccer plate or plate in which you eat the food and
if that is made as a parabolic shape then that flatter plate will have a larger f/d ratio.

As I said majority of the time you are starting can be 0.5 as f/d. However, it actually depends
upon what kind of a feed we have used. So, we will now see one by one what is the effect of
the feed and what is the effect of the radiation pattern of the feed; which is very very
important. So, choosing all these parameters are not independent, they really depend on each
other, the only thing you can say that diameter finding is relatively easy, if you know the gain
if you know the frequency and if you take efficiency roughly say 0.7 or practically as we will
see 0.6 or 0.5 people also got it.

In fact, recently we bought one commercial antenna a reflector antenna at 2.45gigahertz we

did not see the specs in detail. And finally, once they think arrived at our place we were doing
the testing and we realize the efficiency of that was only 0.3; that means we are talking about
30 percent efficient antenna, so which was really a very poor choice to by that antenna. But
nevertheless we could do some initial testing because of course it did give us gain and it did
give us a directive beam. So, except for that gain reduction we could do other testing with
that particular antenna

But that really means that you have to design properly the antenna. So, what are the different
factors which govern the efficiency? Let us see one by one these things.

(Refer Slide Time: 20:01)

So, now gain and aperture efficiency of parabolic reflector antenna: as I mentioned earlier.
So, gain is nothing but efficiency multiplied by directivity, and directivity is given by

4 πA
2 . And for a parabolic reflector A will be nothing but πd2/4. So, you can see that this
❑
2
πd
d2/4 will get cancelled and this will whole thing will become ( )
❑
multiply by this.

Now there are so many things which are associated with the efficiency, this is known as
aperture efficiency. And aperture efficiency depends upon so many of this efficiency there are
total 6 different efficiency are there. So, let us start one by one; this is a efficiency with suffix
s that is known as spillover efficiency. Then εt or efficiency t that is t stands for taper
efficiency P is phase efficiency and then x is polarisation efficiency; b is blockage efficiency
and then r is random error efficiency.

So, in the beginning we will look at these two in more detail and then we will talk about these
efficiencies later on. So, let us first look at spillover efficiency and taper efficiency.

(Refer Slide Time: 21:33)

Now, spill over and taper efficiency depends very strongly on the radiation pattern of the feed
element. Here are two examples. So, this you can see parabola reflector here; this is also
same parabolic reflector. So, these two reflectors are exactly same. We are going to operate at
the same frequency. Now the only difference between these two cases is you can see here,
this is the pattern of the feed you can see that this beam is relatively wider compare to this
beam here. You can see that this beam is relatively narrower.

So, one can just see that reflectors are exactly same, frequency is exactly same, but now the
feed has different pattern. So, let us see what really happen. So, let us start with this particular
thing here. Suppose if you look at this particular feed pattern. So, one can see that the
maximum radiation in this direction which will go the reflector and reflect back. Then from
here this radiation will go here and reflect back, then it can go over here reflect back, and this
part here which is going that will be lost which is also known as spillover.

So, anything which you can see at this particular angle that is nothing but it is spillover. But if
you look at this particular side here now, in this particular case you can actually see that yes
maximum radiation is in this direction which will reflect back this part is same as this side
and, but if you look at this one here at this angle the radiation at this angle is much more
compared to the radiation at this angle. That means there will be a larger spillover here that
means, there will be high spill over here and high spillover means that spillover efficiency is
low. That means, more power is going in this particular direction compare to this particular
case here. So, it has poor spillover efficiency. The other one is a taper efficiency. So, let us
see what is happening over here.

Now, in this particular case we have written here it is high and low, but let us see what is that
really mean. So, in this particular case one can see that radiation is maximum here, but even
if you see at this particular point, if you just imagine a line going from here to this particular
point you can see that the amplitude reduction is relatively small compare to this over here.
At this particular angle you can see that amplitude has reduced drastically.

So now, what is really happening? So this way which is transmitted it goes here reflects back,
it goes over here reflects back. Now you imagine a plane at this particular line here, so you
look at this particular plain here same thing we look at the plain over here. Now, we know
that the property of the parabola is that the point which goes from here comes back or the
thing from here to this particular point, all of them are same distance; so that means the
reflected wave at this particular plane will be in the same phase.

However, this particular thing you can see the reflected wave here, it will have a maximum
amplitude and this amplitude will keep on reducing. But in this case reduction in the
amplitude will be relatively less, whereas in this case we can see reduction in the amplitude
will be much larger because amplitude going over here and reflected back is relatively less.

Now recall the array theory; and in the array theory I had also mention that array theory
becomes the space theory in a sense that we had seen that array factor actually becomes a
space factor for continuous sources. So, I had mentioned that if the number of elements are
increased drastically and the spacing between the elements is reduced drastically then that
would almost become like a continuous source.

Now, if you recall that what we had discussed about the non uniform taper distribution. We
had seen that if there is a uniform distribution that leads to maximum gain, and if there is a
non uniform radiation; that means suppose the radiation is a relatively less; that means from
maxima to this value here it is relatively less or if it is total cosine function which is 0 to
maxima is back to 0. So, we had seen that for a cosine distribution gain was less, for uniform
distribution gain was more.

Or we also discussed about cosine distribution over a pedestal where it at in the end it does
not go to 0, but it goes to relatively lower value. So, you can see between the two cases: one
case is almost becoming closer to cosine distribution and the other case is becoming more
closer to the cosine on a pedestal. So, cosine on a pedestal will have a relatively higher gain
compared to the cosine distribution.

So, that is what the taper efficiency really implies that in this particular case since the field is
relatively uniform or you can say reduction is relatively less, so the reflected wave will have
maxima here and amplitude will decrease slightly. But in this case amplitude will maximum
here, decrees will be much more over here. That means, if this case taper efficiency will be
low, but the spillover efficiency is high.

So, you can see that for the same reflector antenna depending upon the beam pattern in one
case this is high this is low, in other case this is low this is high; That means, if you look at
the product of these two now which sometimes some of the books defined as aperture
efficiency and I just show the plot here quickly.
(Refer Slide Time: 27:59)

So, here is the point this is the spillover and taper efficiency. So, we have shown efficiency
on this side and we have shown reflector aperture angle on this particular side. You can see
that this is the angle psi here which is actually denoting the half power beam width of the
feed horn, and here we have different plots here. These are the plots for half power beam
width of the antenna. So, in one case it will 35 degree then 42.5 and 50 degree. Along this
angle this shows angle θ0. Now θ0 is varying from 0 10 to 90 degree and these are the plots
over here. One can see this is the spill over efficiency plot and this is the taper efficiency plot
over here. And the product of these two in some places they actually say that aperture
efficiency εAP is nothing but equal to εs multiplied by εt, and sometimes that tend to ignore
the other four efficiencies but that cannot be ignored, but right now let us just look at the
product of the two.

So, if you look at the product of two: this is one this is close to 0 you can see that the
efficiency will be very very small. And over here you can see that the efficiency will be
relatively small somewhere in this area efficiency will be maximum.
(Refer Slide Time: 29:30)

So, I just show you the efficiency curve here. So, even though it says aperture efficiency
please remember, this is only two efficiencies we are talking about εs multiplied by εt. So,
one can see that this aperture efficiency is close to 0 here, if the reflector angle is just close to
0 and efficiency relatively you can see is somewhere here this kind of flattened which is
actually nice for the antenna designer engineer. And you can see that this number is
somewhere if I draw the line around 70 percent, you can see that the efficiency is of the order
of 70 percent to about 75 percent.

So, we will continue from here in the next lecture. Just to summarise: today we started
discussing about the parabolic reflector and so far we have talked about prime focus reflector,
they have many other types of reflector antenna which we will discuss in the next lecture.
And then we actually looked into a very simple concept that this is nothing but OP + PQ
should be equal to constant, and from there we drive the relation between f/d and θ0. And θ0
is very very important and related to f/d that depends upon the radiation pattern of the feed.
And we saw that the gain is primarily determined by the diameter of the parabolic reflector
which is but multiplied by efficiency. And efficiency intern depends upon θ0 and θ0 intern
depends upon f/d.

So, we will see all of these things in the next lecture. And in the next lecture we will try to
conclude the reflector antenna and also that will be the conclusion of this particular course
also. With that thank you very much and we will see you next time, bye.