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- My name's Frank Sanders. I'm the head
of the Telecommunications Theory division
at the Institute for Telecommunication Sciences,
a laboratory of
the National Telecommunications and Information Administration,
NTIA, located in Boulder, Colorado.
Over the past 30 years, we have developed a series of techniques
for measuring emissions from various types of radio emitters.
In particular, we've developed techniques for measuring
emissions from radar transmitters.
In this series of talks,
I will begin by describing the fundamentals
of RF measurement techniques.
And I will culminate the series by describing our techniques
for measuring emissions from radar systems,
techniques that, hopefully, you can implement yourself.
Accompanying this series of talks are notes
available on our website, and also NTIA reports
that you may find useful.
We hope that you will enjoy watching
this series of videos as much as we've enjoyed producing them.
Welcome to the 20th and last seminar in the ITS
RF Measurement series.
The topic of today's talk is the calculation of antenna
characteristics, meaning antenna directivity
and antenna gain, from the configurations and dimensions
of antennas.
This is hopefully going to be a relatively short talk.
I'm not going to use a lot of math in this talk, but I do
want to use some mathematical results so that an engineer
who either needs to identify a particular antenna
configuration for a particular task or an engineer who sees
an antenna mounted on a ship or mounted on a building,
mounted on a hilltop, and needs to try to infer from
the antenna dimensions what the characteristics might be
in terms of the antenna gain and directivity in space could
do so based upon basically seeing the antenna
configuration, having some idea of what frequency it
might be working at, and, well, leave it at that.
So some of the material that we're gonna cover right now,
I've talked about in some of the previous talks, and just
as an opener, several things about antennas.
First of all, in terms of functionality...
Ha! I'm not starting out--There we go.
In terms of fun--Sorry.
I haven't had all my coffee.
In terms of functionality, we have two major functions.
One is that antennas, which of course are coupling energy
in and out of circuits via free space, have to somehow
match the impedance of a circuit, and typical circuit
impedances are going to be 50 ohms, 75 ohms, or 300 ohms.
And there are others, obviously, but...Has to match
the impedance of a circuit to the impedance of free space.
And what's the impedance of free space?
Anybody?
- 377 ohms.
- 377 ohms.
So one way or the other, some kind of matching has to
occur between the impedance of a circuit and the impedance
of free space.
The second thing that antennas do is to provide some sort
of geometric directivity to the energy, which is either
being transmitted into space or being received from space.
Now, the directivity could be essentially 0 directivity,
in which case we have an isotropic antenna that
radiates and receives equally in all directions in space.
But these are the two major things that are going to have
to be accomplished by any particular antenna design.
Now, basic antenna types.
These are my breakdowns on antennas.
Basically 3, and the third type sort of breaks out into
a little bit of a 3a down here.
but the first type, which I'm not gonna talk about at all
today, is an antenna that typically works at low
frequencies which couples energy between free space
and a circuit via the B-field component in any
electromagnetic wave.
And these are typically but not always basically toroids
that have some kind of a winding on them, and these
types of antennas are going to typically be used
at low frequencies.
We've used antennas like this to work down at frequencies
of a few hertz or maybe even a few kilohertz.
Second antenna type is an antenna which is a design that
is based on tuned elements, and I'll put that
in parentheses.
It could be a single element or multiple elements.
These antennas include but are not limited to dipole
antennas, stacked dipole antennas...
Oh, yeah, and Yagis, which are narrowband antenna designs,
as are dipoles--narrowband--
and stacked dipoles--narrowband.
They're inherently narrowband because they're built out
of one or more elements which are in fact tuned to the
wavelength of the frequency that they are supposed to
interact with.
And then finally, log-periodics.
Log-periodic antennas.
And these are by definition wideband, and the reason that
they're wideband is unlike a Yagi antenna, which, if you
see, it will look this, seen from the top or seen
from the side...
in which the--in which the element size
and the spacing between the elements, which I've tried to
draw as a constant, is adjusted for a single
frequency, with a log-periodic, we have an antenna
design which consists of a set of elements that vary in size
down the length of the antenna.
So here we have a group of elements that are tuned to
a variety of frequencies and hence is wideband.
Now, in contrast to the tuned element designs, we have
antennas that consist of physical apertures.
And these are the ones that we're gonna be most concerned
with in this particular talk, and these include antennas that
use parabolic--
dish, that is--reflectors, which focus energy on a point
in space where we place an antenna that actually
receives energy.
I'll draw a picture of one of these in a moment.
And also horn antennas.
And these antennas have non-zero--They have non-zero
physical apertures that are actually capturing energy
from space.
And finally--I'm gonna draw it like this.
I'll make it 3a.
We have phased array antennas.
I'll call them arrays, or phased arrays.
Phased arrays.
And these can be a bit of a cross between the tuned
element designs and the physical apertures in that
in a phased array antenna, which I'm gonna draw as
a plate with a lot of elements on it,
we could, for example, mount a whole set of dipole
elements, like so.
And the antenna functions--we talked about this in previous
talks--by phasing the energy from one element to the next
such as to produce a planar wavefront at a distance out
in front of the antenna which can be directed
in space electronically.
Or we can actually build an antenna out of little slots,
and I'm intentionally drawing these at an angle because how
we design it is really completely arbitrary
in many respects.
But we can design an array of slots.
And in any event, to some extent, this array works like
a physical array.
It does have a nice finite size that captures energy,
but on the other hand, functionally speaking,
it's working as an array of individual elements which may
well actually be basically tuned elements.
For example, I've seen some large phased array antennas
on the PAVE PAWS radar which we've seen in one
of the previous talks.
It's a 10-story-tall building which has about 1,700 crossed
dipoles mounted on the face.
So...
Leave it at that for the moment.
One of the things I would like to distinguish momentarily
between antenna versus reflector is on a parabola--
and we often talk about a parabolic antenna or
an antenna with a parabolic reflector, and of course,
we can have the reflectors which are not necessarily
parabolas, but which instead are shaped somewhat like this
and have a reflecting surface like so,
which I'll elaborate on in a moment.
What we typically have on the antenna is a reflector,
which I'm going to draw in 3 dimensions like this.
So here's the side or the back of the reflector coming back
in like this.
So we have a dish scooped out here.
And this is technically not the antenna.
This is the reflector.
The antenna, to be more correct or more precise,
is going to be an element or a set of elements which,
for example, could be a log-periodic array which is
mounted here mechanically, like so,
at the so-called feed point, and this thing is actually
mounted structurally onto the reflector like so.
And the concept, to draw this antenna in cross-section,
is as follows.
Here's the parabola seen with exaggerated curvature
from the side.
And rays that enter the parabola parallel to each
other are reflected back up to a common point in space,
and that point in space is where we mount an antenna
which could, if we wanted this to be a wideband system,
for example, be a log-periodic array like so.
And then the energy is coupled from the dish down into
a circuit of some sort.
So technically, this is the reflector.
Technically, that's the antenna, but oftentimes people
simply refer to these as parabolic dish antennas.
Same thing for an array like this.
Here, the feed gets mounted out in front of the antenna
reflector like so.
And you see these on the FAA air traffic control radars.
We've seen these, for example, on the ASRs in the earlier
slideshows in this series, but the concept is the same.
The energy is focused down to this point where an antenna
is located.
However, typically on these antennas, and also sometimes
on the parabolic dish antennas, we don't necessarily
use a log-periodic.
We could use instead a horn configuration, and this segues
nicely into the next thing that I'll mention.
Try to draw this a little bit in 3-D here like so.
Here's our horn coming back like that.
And again, the horn will feed down into the actual
circuit where a receiver or a transmitter is located.
But the horn, of course, is just that.
It is typically a rectangular aperture.
Seen in cross-section, it'll look like this.
And it is typically but not always built in a flared
configuration like this, like so.
Try to draw in 3 dimensions here.
And I'll talk more about this in a moment.
But both the horns and the parabolas have physical
capture areas that can and do intercept energy traveling
through space in this physical aperture.
So...
Now, I mentioned that in addition to matching the
impedance between free space and a circuit, antennas
have directivity.
In an earlier talk in this series, we discussed the case
in which the directivity of an antenna or of any radiation
space is completely equal in all directions.
This is isotropic, like so.
And here I'm going to draw the surface of a sphere like this,
and we have equal directivity in all directions in space
in and out of the sphere.
Isotropic.
I do not believe that any isotropic antenna has ever
been physically realized as such.
It has occurred to me that if we heat a piece of metal to
a high temperature and let it radiate that it will radiate
isotropically, but in terms of building a practical antenna,
the problem with building something that would be truly
isotropic is that somewhere, somehow, you would have to get
energy in and out of the antenna through a circuit
connection, and it would be probably hard to do that
in a pure sense without somehow disturbing the overall
isotropic characteristic of the antenna.
But it is possible to come close.
Most antennas, however, do have a pattern of radiation
reception and transmission into space, which looks
somewhat non-isotropic.
For a dipole, the pattern looks like this, and we can
build the dipole in here like so.
And we've talked in just the last talk about, for example,
the efficacy of using the nulls in a dipole pattern
for direction finding.
Dish antennas and horn antennas--here's a dish--will
have some kind of a directive pattern that looks something
like this.
And inevitably, even though they're not desired, there
will be non-zero side lobe structures.
Here's the main beam.
Same thing for horn antennas.
Here's a horn seen from above or from the side as
the case might be.
And again, the horn is going to have some kind
of directivity in space.
And this is obviously a geometric characteristic of
an antenna as opposed to an electrical characteristic that
we get with the impedance matching.
I mention this because the way that I like to treat antennas
in any event is as apertures that interact with waves.
So apertures interacting with waves.
We imagine--and I'm gonna construct a--what I'm gonna do
here is I'm gonna construct a bit of a theoretical model
for an antenna behavior, and then we'll compare this
to real-world antennas, and we'll finally end up by
calculating the characteristics of the antennas.
Imagine that we build a wall and the wall has a slot in it
like this, and it's a small slot right here, little
narrow slot.
And the rest of this wall is impermeable to radiation.
And on one side of the wall, there is a wave which
propagates toward the wall like so.
And what I'm drawing here are successive wave crests or
wave troughs.
And of course, as the energy emerges from the wall and we
draw the wave crests or wave troughs, the spacing between
the troughs on this side is the same as the spacing
on this side, but what emerges from the wall are a set
of spherically oriented wavefronts.
And of course, this is a Huygens wavefront coming out.
And we can imagine equally well that if we cut a second
slit in the wall right here, then both slits independently
behave the same as a single slit would, and so we get two
sets of Huygens wavelets coming out, like so.
And again, I'm drawing the crests of the wavefronts.
So this is a classic diagram, of course, from
physics textbooks.
And the result is that if we have a screen set up here
at a distance and we plot the intensity of the energy
on the screen, say, for this double-slit system, we'll get
a set of...
We'll get a set of maxima and minima like this, where I'm
plotting on this axis the intensity of the radiation...
that's falling on the screen from the two slits.
And of course, this is a 2-slit diffraction pattern.
2-slit diffraction.
Again, nothing you don't know or haven't seen here,
I will assume.
Now, we assume here that the slit has essentially 0 size,
essentially 0 width, that the size of the slit is small
compared to the wavelength of the radiation.
We know that if we build a physical aperture
with a physically large-- a physically large slit like
this, that we can model it theoretically as a set
of these arbitrarily small slits like I've drawn here
and that if the spacing between these arbitrarily
narrowly spaced theoretical divisions of this physical
slit is labeled as d, then if we blow up this part
of the diagram, what we get is a geometry that looks like
this, where we have wave energy emerging at some angle
theta, like this.
And here's d.
I'm expanding d out.
I'm zooming out on it like this.
We zoom out like this.
So here is d.
And we have energy emerging here like so.
And we'll draw a normal, down like this.
So we get the same angle theta here as we get here.
And we know that we get a path-length difference--
a path-length difference between this point and this point,
between two successive points, which is equal to d--
this spacing that we can make arbitrarily small--d times
the sine of theta.
And it's this path-length difference between adjacent
points in our theoretically constructed slit that results
in the diffraction pattern that's produced.
OK?
So again, so far, so good.
Now, imagine...
Imagine, as the board rolls forward.
Imagine that we treat an antenna as if it were a slit
with a finite size like I just drew which is acting as
an aperture for wave radiation.
Well, it turns out that, number one, that's a pretty
good model, and number two, if we draw diffraction
patterns relative to--and here's how I want to do this.
We imagine that we have a slit here like I drew
on the previous board.
The slit has some size.
It could be a double slit, but we'll draw it like this.
And imagine that we move out in front of this slit,
which we're going to model as our antenna.
Here's our slit, which is gonna be functionally
equivalent to an antenna, because an antenna is
an aperture and it radiates or absorbs wave radiation.
So we put some kind of an antenna in here.
Let it go ahead and radiate like this.
We imagine that we then put screens in front of the slit--
Actually, I'm gonna use one screen.
We put it at some distance in front of the slit which is
equal to r1, and then we move our projection screen out to
a second distance r2, and then we move it out to another
distance r3, and so forth on out.
And we're gonna plot the diffraction pattern that falls
on the slit as we start out close to our antenna and then
as we move our projection screen further and further out
away from the antenna.
And what happens is when we are at a distance r1, which
I'll draw here like this, we will see a diffraction
pattern which looks somewhat like this.
And down.
And this pattern is described mathematically by
Fresnel equations.
And then we move out to a distance r2, and at a distance
r2, we're going to get a similar but different-looking
pattern which is gonna look like this...
and so forth.
And then we move to a further larger distance r3.
We get a pattern that's gonna look something like this.
And finally, as we move out in distance, we're going to find
that these patterns, these projection patterns, these
diffraction patterns that we get from this, our virtual
antenna, are going to approach a pattern that looks
like this.
One big lobe with little, small lobes.
And where all of these patterns up to this point--all
of these patterns that are in fairly close to the antenna--
and I'll define close in a moment--are described by
the Fresnel relationships, which are exact solutions
or equations to this diffraction problem.
We find that as we move out to a large distance--and again,
I'll define this and what this distance is in just a moment--
we find that the diffraction patterns
begin to converge to this
pattern as we move out to arbitrarily large distances.
And this is the solution that occurs as we approach
an infinite distance from the screen, and this is described
by the Fraunhofer approximation to
the Fresnel equation.
And this approximation gets better and better as the
distance from the slit from our so-called antenna begins
to approach infinity.
Now, geometrically what's happening is, as we move
further and further out away from this antenna,
the radiation wavefront is basically looking less
and less spherical and instead is approaching something that
looks like a flat planar wave front.
One way to think about this is radiation is coming from
a distant star.
At the distance between us and, say, Alpha Centauri,
the radiation that is arriving at the Earth from Alpha Centauri
is virtually a flat plane wave.
Now, the question is, how far away do we need to go before
we arrive at this nice, flat planar wave solution
which behaves basically perfectly like this Fraunhofer
approximation to the Fresnel equation, which really
technically only occurs at infinity?
Well, the short answer is that strictly speaking, you never
get there.
Strictly speaking, any distance is less
than infinity.
But as a practical matter, the so-called far-field
approximation for an antenna pattern--because that's what
we're talking about, is the pattern that we converge on as
the final pattern that we see at virtually basically any
distance away from the antenna as we go out to distances
approaching infinity--is going to occur at a distance which
is equal to--and there are two different criteria for this.
If you're more liberal, you'll say that it occurs
at a distance of d squared over lambda, and if you're
more conservative, you'll say that this limit occurs
at 2 d squared over lambda.
Bear in mind that you never really get there.
But d squared over lambda or 2 d squared over lambda,
where lambda is the wavelength of the radiation that is
coming out of the antenna
and d is the antenna's physical aperture diameter.
So d squared over lambda or 2 d squared over lambda.
If you're doing a measurement and you need to get into
the far field of an antenna, you could probably justify
either of these.
Any distance beyond d squared over lambda or 2 d squared over
lambda is probably gonna be accepted by almost anyone as
having been in the far field.
And by the way, the so-called near-field patterns
for an antenna are related to the far-field patterns by
the Fourier transform.
So we can Fourier transform the near field to get to
the far field or inverse Fourier transform to move
between the two.
So technically, if you make a measurement of an antenna
pattern in a near field, you can calculate the far
field or vice versa.
People like to work typically in the far field of antennas
because they can assume that they're getting a plane wave
out in the far field.
Now...
One thing that I think is very significant about the comment
that I just made is that the behavior of an antenna,
geometrically speaking, in terms of the field
that it produces in space,
is a function--the antenna behavior--the antenna
behavior geometrically is a function of--is a function
of the variables of both the aperture size, d;
and the wavelength of the radiation that were working
with, lambda.
Now, one thing that I think is worth remembering is that any
time that radiation with a wavelength lambda, which I'm
gonna arbitrarily draw like this, interacts
with a physical object, which I'm going to draw like this,
if the wavelength of the radiation is commensurate
with or substantially larger than the size of an object
with which the radiation interacts, the object is not
going to produce a strong amount of directionality
of that radiation in space.
Here's an example.
You can still hear me when I turn my back to you.
The reason that you can hear me when I turn my back to you is
that the acoustic waves that my mouth is generating are
substantially longer in wavelength than the size of my
mouth, and so when my mouth is running like this--here are my
lips, sort of Pac-Man lips-- I'm producing this sort
of Huygens wavefront, which actually--and I'll go up
and draw--do a crude drawing of my nose, and here's my eye,
and here's my forehead, and here's my neck coming back
down and around like this.
And here's my chin coming back down around like this.
The energy is actually going to--is actually going to
diffract all the way back around my head.
If, on the other hand, I could produce acoustic waves that
were small in wavelength compared to the size of my
mouth, I could beam my voice out like a microwave beam
coming out of a dish antenna where the parabola would be
substantially larger than the wavelength, and I could
actually beam the radiation at you, and if I were
looking at you, you could hear me, and if I
turned away, you wouldn't be able to hear me.
But to do that, I'd either have to have a much bigger
mouth--it's hard to believe that that could be possible--
or produce waves that were much, much smaller compared to
the size of my mouth in order to beam the energy
around in space.
So another place where we see this phenomenon occur is the
wavelengths of visible light when they interact with water
droplets in the Earth's atmosphere.
Here we have a lot of little water droplets in the Earth's
atmosphere, and these water droplets are actually fairly
small in size compared to the wavelength of visible light.
We covered this in an earlier talk.
Typical wavelength for visible light is, what, 5,000 angstroms,
if we are looking at green light, for example.
The visible spectrum runs from basically, give or take,
around 4,000 angstroms to around, oh, what,
we'll say 6,000 angstroms just in round numbers.
Water droplets have size that are small compared to the
wavelength of visible light.
As a result, if we have a lot of these droplets arranged
in a cloud in the Earth's atmosphere, what happens is
all of these water droplets diffract the energy
in a highly nondirectional way, and the result is that we
see the energy scattered--and I'm gonna do a crude drawing
of a cloud in the sky--well, not that crude.
Let me draw a thunderhead.
Here's a thunder cloud seen in cross-section
on a hot summer day.
And we see this thing as appearing white because it
consists of water droplets that are scattering all
of the energy in visible light at all frequencies in visible
light basically equally in all directions, and the result is
that we're seeing white.
And of course, the reason that the sky is blue is
because the oxygen molecules, which, again, are much smaller
than the wavelength of visible light
are scattering light via Rayleigh scattering,
and the blue light is more strongly scattered,
the short wavelengths are more strongly
scattered by the diatomic oxygen in the Earth's
atmosphere than the longer wavelengths, and so we have
Rayleigh scattering, which makes the sky blue.
So there, now you've heard the explanation for why is
the sky blue?
But in all these cases, the energy is being scattered
by and is interacting with elements that are much,
much smaller--my mouth, water molecules, diatomic
oxygen--then the wavelength of the radiation itself.
The situation changes when radiation interacts
with objects that are somewhat larger than the wavelength.
When energy is interacting with objects that are
substantially larger than its wavelength, then the object,
whether it's a parabolic reflector or some kind
of geometrically constructed device, becomes capable
of strongly directing energy in space.
So here we'll draw a parabolic reflector like this, and we'll
draw the wavelength of the radiation that's going to
interact with it like that.
So here's this little, tiny lambda down here
interacting with this rather large reflector.
When this occurs, geometrically it's possible
for this reflector to produce a fairly strongly directional
beam in space.
It's really as simple as that.
Now, it's interesting, though, and informative to have some
idea of what these wavelengths are for
electromagnetic radiation.
So I'll do a little drawing here, drawing a graph--
not a graph, a table.
Excuse me.
We'll do frequency, and then we'll do lambda.
And we'll do this for 1 MHz,
10 MHz, 100 MHz, 1,000 MHz--1 GHz--
and finally we'll just do 10--10 GHz--like so.
We want to know this because we want to know what size
antenna we would need to build in order to produce fairly
strongly directional beams for any particular frequency that we
might ever need to work at.
So the wavelength lambda is going to be equal to the speed
of light divided by the frequency, nu in this case.
Speed of light is 3 times 10 to the eighth meters per second,
so 1 MHz is going to be 300 m in wavelength, and then I
intentionally did this as a log progression.
30 m, 3 m,
30 cm--basically about 1 foot at this point--
and finally 3 cm.
And based on the description that I just gave, if we want
to direct this energy strongly in space, we're either going
to need to build a parabolic reflector at any given
frequency which is substantially larger than this
wavelength lambda, or equivalently, we may want
to build some kind of a phased array, but even if we want
to do that, the phased array size overall
is going to have to be-- and I'm drawing a phased array
with elements on the phase-- the phased array size is
still going to need to be rather large compared to
the wavelength of the radiation that we are working with.
So suppose that we want to build a space search radar
like PAVE PAWS.
We decide that 400 MHz frequency, which is between
here and here, is a good frequency to use in terms
of propagating through the Earth's atmosphere
and interacting with objects in space.
But we needed the big antenna that were gonna build
at 400 MHz to be substantially larger than a wavelength.
Well, that means that it's gonna have to be substantially
larger than at least a few meters.
And in fact, we know that the PAVE PAWS radar--we've seen
photographs of it earlier in the series--is a building that
is 10 stories tall.
Basically, it has to be a large building because it's
working at a relatively low frequency.
On the other hand, we say that, for example, we need to
operate an airborne radar.
This airborne radar needs to have a highly directional
beam, and it needs to fit inside the nose of an airplane or
inside the belly of an airplane.
Is this going to work well if we try to build a radar
down at 1 MHz?
No. Lambda is too large.
Airplanes aren't 300 m long.
Airplane antennas that go inside airplanes aren't going
to be 30 m in size.
Even 3 m in size is a push, although Navy AWACS planes
work with radars at about 400 MHz.
But more practically, these are the wavelengths where we
can build fairly compact antennas that are well adapted
for fitting inside aircraft, spacecraft, small boats,
small vessels.
And aside from propagation characteristics, that's why
these frequencies from 1,000 MHz upward to 10 GHz and above
are considered highly desirable for radar systems.
It's because we can build relatively small antennas
which are nevertheless large compared to a wavelength
of the radiation that they are interacting with.
So, again, it's basically as simple as that.
The other thing that we can do is to use this information to
look at a dipole antenna and tell by looking at the dipole
antenna, assuming it's a half wave dipole, roughly what
frequency it might be working at, because, just to expand
part of this table a little bit, we can look at
300 MHz frequency, 400 MHz, say 800 MHz.
And what is lambda and what is lambda over 2
at these frequencies?
We'll say that we are looking at a Yagi or we need to build
a Yagi antenna or a dipole that's gonna work
at these frequencies.
And we get a wavelength of 1 m,
75 cm, and basically 38 cm,
and if we divide these numbers by 2,
we get half a meter, we get about 38 cm, and finally we
get about 18 cm here just in round numbers.
So if we are looking at either a conventional dipole antenna
or a Yagi antenna that has a set of elements mounted on it,
we can quickly estimate what frequency it must be operating
at just by looking at the size of the antenna.
Again, conversely, if we need to build a Yagi to work
at these frequencies, we know immediately roughly how large
the--how large, how long the elements are gonna need to be
because these elements are in fact tuned elements tuned to
these frequencies, tuned to these wavelengths.
So...fairly straightforward.
An easy way to remember all that is 300 MHz is gonna
produce a 1 m wavelength because the speed of light is
3 times 10 to the 8 m per second.
Now...
Believe it or not, we are actually close to being done.
I want to talk about directivity, which I'll
write as d, versus gain, g.
Directivity is equal to-- the directivity of an antenna.
If we're talking about beaming energy into space, receiving
energy from space, the directivity of an antenna is
gonna be equal to the power density in some lobe or beam
coming out of that antenna divided by the power density
that would have been produced had we been using
an isotropic antenna.
So it's the difference between what the antenna is actually
putting into space in a direction versus
what the antenna would've been putting into space if it
had radiated energy equally in all directions in space over
the surface of a big sphere.
And...I'll leave it at that.
Now, gain is equal to the directivity of an antenna
multiplied by the actual electrical efficiency
of the antenna in terms of energy that's coupled from
a circuit.
This is dependent entirely on energy in space divided by the
energy that would've occurred in space due to an isotropic
radiation pattern.
Directivity.
Gain is directivity multiplied by whatever the efficiency
of the antenna is in terms of coupling energy in and out
of a circuit.
This is a more practical number to use
because generally what happens is if we hook up an antenna
and we run it into some kind of a circuit over here and we
are measuring power coming in or out of the antenna,
we're gonna do it in or out of the circuit with the circuit
hooked up to some kind of a power meter, which
in principle has some kind of a needle that shows this
amount of power in a circuit like so.
So it's as simple as that.
So gain is a geometric effect.
It's related to directivity,
but technically we have this little factor, a, to put in.
Now, the definition of the effective aperture, a sub e--
this is definitional.
There are justifications for the definition.
I'm not going to go into them here.
But by definition, the effective aperture
of an isotropic antenna is equal to the wavelength
of the radiation that's coming in or out of it square,
lambda squared, times g
divided by 4 pi.
4 pi is, again, a geometrical feature related
to the fact that there are 4 pi steradians
in a sphere or in a spherical distribution--
in a distribution of--well, I was gonna say spherical
distribution of energy,
but really, a spherical distribution of anything.
There are 4 pi steradians in a sphere, period.
OK, so lambda squared times g over 4 pi.
Now, this is definitional.
What this says is--well, I'll leave it at that.
But suppose that we want to represent the effective
aperture of an isotropic antenna in a propagation
calculation--we've done this in an earlier talk
in the series--such that we're gonna work in decibels.
And so, at some point in our
calculation, we have to take 10 log of a sub e.
Well, that's going to be equal to 10 log of lambda squared,
which is 20 log lambda,
plus 10 log of the linear gain
minus 10 log of 4 pi.
Now, rather than work in terms of wavelength, we typically
like to work in terms of frequency.
And remember that lambda is equal to c divided by nu,
or here I'm just gonna write f for frequency, and so this
means that the effective aperture in decibels written
in capital letters, which is equal to 10 log A sub E,
is equal to 20 log of the speed of light--so that's
a constant--minus--this is important--minus 20 log f
plus 10 log of little g, which is big G.
The convention that I've always used is that any
parameter appearing in capital letters is a decibel quantity.
If it's in lowercase, it's a linear quantity.
Minus 10 log of 4 pi.
And here's where I'm going with this.
Here's where I'm going with this.
This is a constant. This term is a constant.
And so that means that we have some constant terms that we
pull together.
Then we'll go ahead and add big G,
and this is gain in decibels relative to isotropic.
People often write the gain of an antenna as just dB, but it
is--it's usually decibels relative to isotropic.
Now, if they are comparing it to a real engineer-constructed
antenna, then they are typically comparing it to
a dipole, in which case it'll be decibels relative
to a dipole, which is dBd, but we'll leave it as dBi here.
And then, and this is the tricky part I want to come to,
minus 20 log of the frequency.
So the functional dependence for the effective aperture
of an isotropic antenna, which we're never going to
actually build, is going to be some kind of a constant plus
it's gain in dBi--and by the way, we can relate this to
the gain of a physical antenna, a dipole, which we would write
as decibels relative to a dipole, by a factor of 2.1 dB.
A dipole should have a gain of 2.1 dB over isotropic.
This term minus 20 log f.
This term for the effective aperture of an isotropic
antenna, or equivalently, even for a dipole gain,
when it appears in calculations related to
propagation, will show up in the calculation.
And I have on numerous occasions heard people say,
"Oh, we don't want to work at 9 GHz,
"or we don't work at 5 GHz, because we don't want to have
to deal with all that loss at those high frequencies."
This equation has nothing to do with loss
at high frequencies.
This has to do with the definition of the effective
aperture of an isotropic antenna, which is decreasing
as frequency increases, and there is no relationship
between this and the loss through space.
Space always results in the decrease of the intensity
of a wave as a function of the distance from the source
of the radiation squared, period.
It's geometrical.
Space doesn't know anything about frequencies.
So when you see this, if you ever see it, please don't
tell me that you don't want to work at 9 GHz because of that
20 log f loss that you get at 9 GHz.
It has nothing to do with 9 GHz.
Has nothing to do with the antenna.
In fact, as we'll see in a moment, the effective aperture
of a dish antenna goes up at the rate of 20 log relative to
isotropic, which means that it's flat,
because it's the effective aperture
of an isotropic antenna,
which is actually dropping as 20 log of the frequency.
OK.
So...just like to point out that.
So...We imagine that we're going to build
a directional antenna.
It's going to radiate energy in some sector in space.
I draw a sphere here.
And then on this sphere, I draw a little rectangularly
shaped zone which measures 1 degree by 1 degree, like so.
And how many of these little 1-degree squares fit
onto the surface of a sphere?
And the answer is 41,000-- check my numbers.
Yeah, 41,253 of these little 1-degree-sized areas
on the surface of a sphere.
So if we can collect--if we can collect all of the energy
coming out of the radiator here and direct all of that
energy into this 1-degree zone, like so, radiating out
into space, then we will have concentrated the energy coming
out of that radiator relative to having distributed it over
the surface of a sphere by a factor of 41,253.
So the gain of that antenna would be equal to 41,253
divided by 1.
If you will, it's a sort of an amplification factor in terms
of squeezing the energy down onto a single azimuth.
And 10 log of 41,253 would give us the gain of this antenna
in decibels, and that is, I think, 46 dB.
I'm checking my notes.
Yes, 46 dB, like so.
So a geometric gain factor of 46 dB over isotropic means
that we are squeezing all the energy down into one square
degree in space.
That's kind of an interesting or useful mnemonic.
Now, if we more accurately model this not as
a rectangular zone, but as a zone with a little circular
cross-section like so, then the number changes
slightly, and what we get is-- and what we get is that g is
equal to 41,253 times 4 over pi,
and that's equal to 52,525--5-2-5-2-5.
And 10 log of that number,
which I think I did.
I didn't actually do it.
It's gonna be just slightly larger.
It's gonna be about 47 dB.
So it doesn't make a huge amount of difference,
but, again, if we can squeeze the energy down to one square
degree, we get about that.
Now, real antennas are not able to squeeze all the energy
into a single zone like this with no other energy radiated
in any other direction.
I've already alluded to the fact that the real radiation
pattern of an antenna is going to look something like this.
Typically the first side lobe on either side is going to be
the highest, but that can vary.
But we do have energy coming out in these other lobes,
and this represents a loss in efficiency for the antenna.
And a typical realistic efficiency for an antenna,
realistically, compared to an antenna that can theoretically
perfectly guide and direct all the energy in space,
is about 55%.
So now we can put--believe it or not,
we can put the whole picture together.
We understand the relationship between directivity and gain.
We know that a typical efficiency--I'm just going to
state it, I'm not going to prove it--a typical achievable
efficiency is about 55%.
In other words, of all the energy that this radiator is
producing, about 55% of it is coming out in this main beam,
and the remaining 45% is distributed all the way
around here in the side lobes and back lobes.
So what we can do is actually draw up a little table where
we write down the aperture of the antenna.
I'm gonna go ahead and erase this.
And then I'm going to write the beamwidth as a function
of the aperture-- of the aperture.
And we're going to do all of our calculations
in natural units.
We're not going to use meters or centimeters.
We're gonna do all the calculations
in terms of wavelength.
The wavelength lambda, whatever wavelength we pick,
is going to be our unit of measure
for these calculations.
And then we have directive gain as a function...
of the aperture.
And then lastly, we have the gain itself, the gain,
as a function of the beamwidth.
And I'm gonna do this for the two most basic physical
apertures that we will ever encounter: a horn aperture--
a rectangular aperture on the end of a horn--and a circular
aperture of a parabolic reflector.
I'll do it like this.
Gonna need a little space here, as we'll see
in a moment.
And so.
And then we are going to divide it right down
the middle for, first of all, a horn,
and we've got two dimensions that we're gonna
consider for this horn.
I'm gonna draw the horn twice.
We're gonna consider the effective aperture of what
we'll call--well, not call, but what is the E field
interaction for the horn-- we'll call it vertical--
and the effective aperture for the H field for the horn.
This will be the horizontal dimension on this thing.
And for the parabola, we'll look at it face on, and we'll
just say that it has an aperture, a.
And so, every beam that we draw--every
beam that we draw--has two dimensions, an x and a y,
and we're just gonna write these as theta 1, say,
for the x, and theta 2 for the y.
These are angular beam dimensions.
If the beam has a circular cross-section, then theta 1
is equal to theta 2, but it doesn't have to have
a circular cross-section.
And theta 1 doesn't have to equal theta 2.
But the area of the beam is basically gonna be the product
of these two.
So theta 1--and we could go through the math on this,
but we're not going to--theta one is equal to roughly--Oh.
At 55%--at 55% to 60% efficiency.
So this is for real antennas, real live antennas
with 55% to 60% efficiency.
Theta is equal to 67 lambda over a sub E
and a sub H respectively.
And down here, we'll say that theta 1 and theta 2 are
in fact equal to each other and they're both equal to
72 times lambda divided by the diameter a.
We're just assuming symmetry on the dish here.
Divide by the diameter a.
Again, that's for real-world efficiency.
And so this means that, as it turns out, the gain is
equal to 7.5 times a sub E times a sub H
divided by lambda.
That's linear gain.
That means that the decibel gain is equal to some constant--
we just take 10 log of g-- it's gonna be equal
to some constant plus 10 log of a sub E
plus 10 log of a sub H
plus 10 log of f.
And finally, if we want to look at the gain as a function
of these beamwidths, it turns out it's equal to about 31,000
divided by the product of the beamwidths theta 1
and theta 2.
And finally, for the dish, the linear gain is equal to about--
these are all "about" because these are
approximations for this efficiency--
5 a squared over lambda squared.
And that means that the decibel gain is equal to
a constant--again, we take 10 log of little g--
a constant plus 20 log a plus--
yeah, 20 log a plus 20 log f.
Remember f goes inversely as lambda.
It would be minus 20 log of lambda.
So it becomes--yeah, so it becomes plus 20 log f squared.
Sorry, excuse me, squared.
Lambda squared here.
5 a squared over lambda squared.
So the lambda squared goes to 20 log f.
And look at that.
Remember that the effective aperture
of the isotropic antenna dropped as minus 20 log f?
Well, the gain of the dish relative to isotropic goes up
as 20 log f.
In other words, in real terms, in terms of the aperture being
presented, it's a constant aperture antenna.
That's consistent with the fact that in fact,
the aperture of this dish is constant.
In other words, this flattens out the minus 20 log f
that we see come out of the calculation
for the effective aperture of an isotropic antenna.
And functionally, the linear gain of the antenna is
about 27,000 divided by--
assuming that these two are equal to each other--
divided by the beamwidth theta squared.
Simple as that.
And that basically concludes the talk.
Anytime that you see an antenna, either a horn antenna
or a parabolic dish antenna,
or even a radar antenna that's a sort of a cosecant squared,
you can roughly calculate
the gain if you know what wavelength it's working at,
or you can work the relationship backward if you know
the wavelength but you don't know the gain.
You need to know one or the other, but if you know one or
the other, you can back-calculate the other one.
This also explains, again, why we don't typically build
dish antennas to operate at 10 MHz.
We do have dish antennas at the Department of Commerce
Table Mountain field site which will function
at frequencies of a few hundred MHz, and those
antennas are 60 feet in diameter.
I hesitate because I think it's 60 m, but they are
about 60 feet in diameter.
So that's probably about the largest-sized dish that can be
practically built as a steerable dish.
The Arecibo dish in Puerto Rico, which is not steerable,
could work at even lower frequencies.
But basically, these are the relationships between
beamwidth and gain as a function of aperture and gain
as a function of beamwidth.
So with this, as I say, you should be able to either
calculate the size antenna you need for a given application,
or if you're working with an established antenna, you can
establish what the gain or the wavelength is that the antenna
is using if you know one of those parameters or the other.
Any questions?
Well, thanks for bearing with me.
Hopefully some of this will come in handy at some point
or another in your career.
Thanks very much.
[Applause]