SEC.2.5]
BEAMS
expresses well enough
OF SPECIAL
for our purposes
the effect of viewing the echoing
(In Fig. 22 the vertical angles
area obliquely,
at least when 0 is small.
have been exaggerated, for clarity.)
Substitution
of this expression
replacement, of R
by
h/sin
s
If S is to
be
antenna pattern
vary
for a in the radar
0, h being
= P =
(4r)'R4
independent
require that G(o)
of
equation,
and
the height
of the aircraft,
= p~2&;&4
'.
0 between
aa CSC20 through
then would
25
SHAPES
O=
the
leads to
(6b)
00 and
8 = 7r/2, we
this angular interval.
be described
must
The ideal
by
for 00<0<
G = Go=&
~/2,
(7)
G = O, at all other angles.
}
The requirement
of Eq. (7) imposes a restriction
on the maximum
gain GOwhich can be achieved and hence on the maximum range.
To see
this, let
us
compute
the
average
directions, or l/47rJJG d~ extended
element of solid
equal to
1.
angle.
But
By
of
the
the definition
first something
must
antenna pattern in the other plane--that
beam.
Suppose
that
the horizontal
is uniform
plane such
horizontally,
as ACEF,
of
the
antenna
of gain this integral
be
over
all
said
about
must
the shape
be
of the
is, in a plane normal to the fan
width
been fixed by other design considerations
gain possible will be obtained
gain
over the whole sphere, where d~ is the
of the antenna
at the value a.
if the illumination
aperture
has
The maximum
of the antenna
and in this case it can be
Fig. 2"2, the gain as a function
aperture
shown that in
of the angle
any
@
will bel
sin2
G+ = G(~=o)
7ra+
—A
2"
rra+
()
(8)
(-) x
As a function
of 8 and @ then, the pattern
is completely
CSC28
G=
GosO
described
by
'in2
()'+4
(-)
(9)
7ra+ 2'
A
2.
and we have to require
L
47r
H
o
.
+2
G cos @d@ dtl = 1.
_;
1This expressionwill,be recognizedas the diffraction pattern of a rectangular
aperture. SeeVol. 12, Sec. 49.
Since G is very
26
THE
RADAR
[SEC.2.5
EQUATION
small except for small values of O, and since G vanishes for values of o
outside the interval 00to ~/2, it is permissible to write and evaluate the
integral
as follows;
=/2
G,
1
4UCSCZ00
-
H
~.
7raij
—
i
()
sin2
CSC2e
o
d@dO=~cos
O,sin OO= 1.
(10)
7ra4 2
(-) A
Hence
GO = ~sec
Returning
Eq.
(n),
nowto
Eq.
andareplaced
OOCSC00,
(6b), if Gisreplaced
byk/a,
S = PLah
When O,issmall,
by Rti,
side of
tanz 0,
4uh3
thecase,
asisusually
the maximum
bytheright-hand
weobtain,forS,
(12)
toreplace
itispermissible
range, which leads tothefinalr
h/tan
00
elation
(13)
The
appearance
of Xin
obtained
forthesimple
theresult
be traced to the influence
cross section
other
parameters,
hR2 is fixed.
That
proportional
to
everything
the
pattern of the antenna,
shape for each height.
A problem
at
altitude
all
h.
and
a,
root
of
the
radar
set
constant
which
we
assume
Here,
however,
verify
point
that
Eq.
targets
ale
upon the effective
that
specified,
once
the
range obtainable
height
of
but
the
to
be
with
(5), can
the
the
quantity
is inversely
aircraft,
vertical
adjusted
keeping
radiation
to
optimum
one is met in the design of groundto provide
(aircraft)
u is generally
this assumption
uniform
flying
assumed
at
to be
cover-
some
limiting
constant:
the
leads again to the require-
as CSC20, but with a diflerent final result for the
of S upon h, R,
target with u independent
be contrasted
beamwidth
radar, which may be required
ment that the gain vary
than hR5.,,
L!&,
square
for
reader will easily
dependence
P,
ranges
isto
It will be observed
related to the preceding
based air-search
age
target.
is to say, the maximum
the
about
which
fan beam and point target,
of the horizontal
of the extended
system
this formula,
and a.
It turns out in fact that for the point
of angle, the quantity
and S is proportional
hR~,.
is constant,
to az rather than to ah.
rather
TIIE
SEC.26]
BEA6'ON
In practice it is not feasible
the specification
achieved
problem
in
of
of
Eq.
the "two-way"
radar
problem,
round
some remote object.
Radar
at a remote
from
transmitter.
the
transmission
in which
as
trip involving
this
point
picks
another
is
signal,
in
reply,
for
equation
by
the
ways
which
consists
simply
Consider the first process,
of
usually
the
called
by
radar,
~b the
beacon antenna.
These must be related by
transmission
signal
equation with
the
same antenna
or
process of beacon
subscripts
similar
power
reception at the beacon
case, the quantity
are
factor
to
long beacon
to
other
hand,
that
achieve
at
in area when
available
decreasing wavelength.
If
transmission
the
shorter
the
aperture
in Eq.
power
has
antenna.
This
be, in general, more
Antenna
apertures
are employed:
generally
this
cover-
(14) is the quantity
of the radar
ranges should
wavelengths.
Actually
is fixed at a relatively
like omnidirectional
shorter wavelengths
r-f
the
and
(15)
in the parenthesis
leads one to suspect
rarely increased
respec-
by a similar
throughout.
for
antenna
of something
G,k*which is proportional
difficult
antennas
P,
the gain G, of the beacon
The remaining
the
relation
s
E-P,
age.
be the
by
has the same value for interrogation
,b
In practice,
be
reception.
received
reply is described
employed
and reply, and we can infer the corollary
low number by the requirement
will
processes,
and
and likewise at the radar, as is nearly always
in the parenthesis
an
Here
(14)
b and r interchanged
antennas
to
provides
8).
Let P,
where G, and Gb are the gains of the radar and beacon
The subsequent
out
()=)
A%=p,
tively.
by
operation
"interrogation."
power transmitted
to
associated
radar,
(Chap.
radar-beacon
one-way
been
to
enabled
an
been
the
A
sent
and
It is clear that we have to do with two entirely independent
each of
has
transmitter
energy
amplified
the analogue
discussed.
radar
of
of the energy
up directly
signal
type
has
on a different principle.
back
at
in various
the
meets
our attention
from
scattering
operate
The
of
the route
transmitter.
This
signal,
received
artificial echo which can be utilized
of
which exactly
antenna,
beacons
receiving antenna
radar
a pattern
a reasonable
approximation
A
comprehensive
discussion
in Chap. 14 of Vol. 12.
Equation.-So
far we have confined
receiver is a complete
initiate
but
several
instances.
the
' cosecant-squared"
called, will be found
2.6. The Beacon
the
to produce
(7)
27
EQUATION
decreases
not
proved
are
on the
markedly
to
be
with
a serious